2.1 Graphs

We use Serre’s convention for graphs [Reference Serre33]. That is, an (undirected) graph is a tuple $G = (V,E,o,\tau ,\bar {\phantom {e}})$ where:

1. (1) V and E are sets, called the vertices and the directed edges (we think of E as containing two copies, with opposite orientations, of each undirected edge);

2. (2) $o,\tau \colon \thinspace E \to V$ are functions that specify the originating and terminating vertices of an edge;

3. (3) $\bar {\phantom {e}} \colon \thinspace E \to E$ is a fixed point free involution such that $o(e) = \tau (\bar {e})$ ( $\bar {\phantom {e}}$ flips edges).

We fix an orientation on G, that is, a subset $E_{+} \subset E$ that contains exactly one edge from each pair $\{e,\bar {e}\}$ . Since we consider the pair $\{e,\bar {e}\}$ to be a single edge, the number of edges of G is $|E_{+}| = |E|/2$ . The valance of a vertex v is the number of edges from $e \in E_+$ with $o(e)=v$ plus the number of edges from $e \in E_+$ with $\tau (e)=v$ (an edge e for which $o(e) = \tau (e) =v$ contributes $2$ to the valance). Oftentimes when defining a graph we only specify the edges in $E_+$ (together with the restrictions of o and $\tau $ to $E_+$ ). The complete set of edges is then defined as $E = E_+ \cup \overline {E}_+$ , where $\overline {E}_+$ is a copy of $E_+$ , and $o,\tau $ , and $\bar {\phantom {e}}$ are defined in the obvious way. We blur the distinction between the tuple $(V,E,o,\tau ,\bar {\phantom {e}})$ and the corresponding one-dimensional CW-complex with $0$ -cells V and $1$ -cells $E_+$ .

The space of length functions on G is the open convex cone

(2.1) $$ \begin{align} \mathcal{M}(G) = \{ \ell \colon\thinspace E_{+} \to \mathbb{R}_{>0} \}. \end{align} $$

We consider this set as a subset of $\mathbb {R}^{\lvert {E_+} \rvert }$ . A length function $\ell \colon \thinspace E_{+} \to \mathbb {R}_{>0}$ extends to a function $\ell \colon \thinspace E \to \mathbb {R}_{>0}$ by $\ell (e) = \ell (\bar {e})$ if $e \notin E_{+}$ . By $\mathbf {1} \in \mathcal {M}(G)$ we denote the constant function with value 1.

An edge path is a sequence of edges $(e_{1},\ldots ,e_{n})$ in E such that $\tau (e_{i}) = o(e_{i+1})$ for $i = 1,\ldots , n-1$ . A function $f \colon \thinspace E \to \mathbb {R}$ (in particular, a length function) extends to a function on edge paths $\gamma = (e_{1},\ldots ,e_{n})$ by

(2.2) $$ \begin{align} f(\gamma) = \sum_{i=1}^{n} f(e_{i}). \end{align} $$

2.2 Outer space

We will introduce some definitions and notation for the Culler– Vogtmann outer space. This space was originally defined by Culler and Vogtmann [Reference Culler and Vogtmann14]. For more information, see for example the survey papers by Vogtmann [Reference Vogtmann35] or Bestvina [Reference Bestvina and Li5].

Let $\mathcal {R}_r$ be the r-rose. That is, $\mathcal {R}_r$ the graph with a unique vertex v and r edges. Fix an isomorphism $\mathbb {F}_r \cong \pi _1(\mathcal {R}_r,v)$ . A marked metric graph (of rank r) is a triple $(G,\rho ,\ell )$ where:

1. (1) G is a finite connected graph without vertices of valence $1$ or $2$ ;

2. (2) $\rho \colon \thinspace \mathcal {R}_r \to G$ is a homotopy equivalence; and

3. (3) $\ell $ is a length function on G.

There is an equivalence relation on the set of marked metric graphs defined by $(G_1,\rho _1,\ell _1) \sim (G_2,\rho _2,\ell _2)$ if there exists a graph automorphism $\alpha \colon \thinspace G_1 \to G_2$ such that $\ell _1 = \ell _2 \circ \alpha $ and such that the following diagram commutes up to homotopy:

We let $\mathcal {X}(\mathbb {F}_r)$ denote the set of equivalence classes of marked metric graphs of rank r. The group $\operatorname {\mathrm {Out}}(\mathbb {F}_r)$ acts on $\mathcal {X}(\mathbb {F}_r)$ on the right by precomposing the marking. Specifically, for any outer automorphism $\phi \in \operatorname {\mathrm {Out}}(\mathbb {F}_r)$ , there is a homotopy equivalence $g_\phi \colon \thinspace \mathcal {R}_r \to \mathcal {R}_r$ that induces $\phi $ on $\pi _1(\mathcal {R}_r)$ via the aforementioned fixed isomorphism $\mathbb {F}_r \cong \pi _1(\mathcal {R}_r,*)$ . Moreover, this homotopy equivalence is unique up to homotopy. With this, we define

(2.3) $$ \begin{align} (G,\rho,\ell) \cdot \phi = (G,\rho \circ g_\phi,\ell). \end{align} $$

This action respects the equivalence relation on marked metric graphs and so defines an action on $\mathcal {X}(\mathbb {F}_r)$ as claimed.

Let $\mathcal {G}_r$ denote the set of finite connected graphs without vertices of valence $1$ or $2$ whose fundamental group has rank r. We observe that this is a finite set. Given a graph $G \in \mathcal {G}_r$ and homotopy equivalence $\rho \colon \thinspace \mathcal {R}_r \to G$ , we set

$$ \begin{align*} \mathcal{X}(G,\rho) = \{ [(G_0,\rho_0,\ell_0)] \in \mathcal{X}(\mathbb{F}_r) \mid G_0 = G \mbox{ and } \rho_0 \simeq \rho \}. \end{align*} $$

There is a bijection $\mathcal {X}(G,\rho ) \to \mathcal {M}(G)$ defined by $[(G_0,\rho _0,\ell _0)] \mapsto \ell _0$ . These sets partition the set $\mathcal {X}(\mathbb {F}_r)$ and are permuted under the action by $\operatorname {\mathrm {Out}}(\mathbb {F}_r)$ . Specifically, for each $G \in \mathcal {G}_r$ we fix a marking $\rho _G \colon \thinspace \mathcal {R}_r \to G$ . Then

$$ \begin{align*} \mathcal{X}(\mathbb{F}_r) = \bigcup_{G \in \mathcal{G}_r} \bigcup_{\phi \in \operatorname{\mathrm{Out}}(\mathbb{F}_r)} \mathcal{X}(G,\rho_G) \cdot \phi. \end{align*} $$

There is a topology on $\mathcal {X}(\mathbb {F}_r)$ that is often defined in three different ways. We will need to use the first two and for completeness we explain all three here.

The weak topology. The notion of a collapse induces a partial order on the set of marked graphs. Specifically, for two graphs G and $G_0$ , we say that G collapses to $G_0$ if there is a surjection $c \colon \thinspace G \to G_0$ such that the image of any edge in G is either a vertex or an edge of $G_0$ and such that $c^{-1}(x)$ is a contractible subgraph of G for each point x of $G_0$ . The map c is a called a collapse. Observe that if the map $c \colon \thinspace G \to G_0$ is a collapse, then a length function $\ell \in \mathcal {M}(G_0)$ can be considered as a degenerate length function $\ell _{G_0}$ on G by

(2.4) $$ \begin{align} \ell_{G_0}(e) = \begin{cases} \ell(c(e)) & \mbox{if}\ c(e)\ \mbox{is an edge in}\ G_0, \\ 0 & \mbox{otherwise}. \end{cases} \end{align} $$

This defines a map $c^* \colon \thinspace \mathcal {M}(G_0) \to \mathbb {R}_{\geq 0}^{\lvert {E_+} \rvert }$ by $c^*(\ell ) = \ell _{G_0}$ . We now define the following subset of $\mathbb {R}_{\geq 0}^{\lvert {E_+} \rvert }$ :

(2.5) $$ \begin{align} \overline{\mathcal{M}}(G) = \bigcup_{c \colon\thinspace G \to G_0} c^*(\mathcal{M}(G_0)). \end{align} $$

We note the $\mathcal {M}(G)$ is a subset of $\overline {\mathcal {M}}(G)$ as the identity map ${\textrm {id}} \colon \thinspace G \to G$ is a collapse.

Next, given two marked graphs $\rho \colon \thinspace \mathcal {R}_r \to G$ and $\rho _0 \colon \thinspace \mathcal {R}_r \to G_0$ , we say that $(G,\rho )$ collapses to $(G_0,\rho _0)$ if there is a collapse $c \colon \thinspace G \to G_0$ such that $\rho _0 \simeq c \circ \rho $ . In this case we write $(G_0,\rho _0) \leq (G_,\rho )$ . We now define the following subset of $\mathcal {X}(\mathbb {F}_r)$ :

(2.6) $$ \begin{align} \overline{\mathcal{X}}(G,\rho) = \bigcup_{(G_0,\rho_0) \leq (G,\rho)} \mathcal{X}(G_0,\rho_0). \end{align} $$

The bijection $\mathcal {X}(G,\rho ) \to \mathcal {M}(G)$ extends in a natural way to a bijection $\overline {\mathcal {X}}(G,\rho ) \to \overline {\mathcal {M}}(G)$ and allows us to consider $\overline {\mathcal {X}}(G,\rho )$ as a subset of $\mathbb {R}_{\geq 0}^{\lvert {E_+} \rvert }$ .

The weak topology is defined using this collection of subsets. Specifically, a set $U \subseteq \mathcal {X}(\mathbb {F}_r)$ is open if $U \cap \overline {\mathcal {X}}(G,\rho )$ is open as a subset of $\mathbb {R}_{\geq 0}^{\lvert {E_+} \rvert }$ for all marked graphs $(G,\rho )$ .

The axes topology. Given a marked metric graph $(G,\rho ,\ell )$ and an element $g \in \mathbb {F}_r$ , we denoted by $\ell ([g])$ the $\ell $ -length of the shortest loop in G representing the conjugacy class $[\rho (g)]$ . This induces a function ${\textrm {Len}} \colon \thinspace \mathcal {X}(\mathbb {F}_r) \to \mathbb {R}_{\geq 0}^{\mathbb {F}_r}$ where ${\textrm {Len}}([(G,\rho ,\ell )]) \colon \thinspace \mathbb {F}_r \to \mathbb {R}_{\geq 0}$ is the function defined by

$$ \begin{align*} {\textrm{Len}}([(G,\rho,\ell)])(g) = \ell([g]). \end{align*} $$

Culler and Morgan proved that the map ${\textrm {Len}}$ is injective [Reference Culler and Morgan13, 3.7 Theorem]. The resulting subspace topology on ${\textrm {Len}}(\mathcal {X}(\mathbb {F}_r)) \subset \mathbb {R}_{\geq 0}^{\mathbb {F}_r}$ is called the axes topology. It is known that this topology agrees with the weak topology. (See [Reference Culler and Vogtmann14, §1.1] or [Reference Guirardel and Levitt21, Proposition 5.4].)

The equivariant Gromov–Hausdorff topology. We will not need this definition, and we only remark that Paulin showed that it is equivalent to the axes topology [Reference Paulin30, Main Theorem].

There is an action of $\mathbb {R}_{>0}$ on $\mathcal {X}(\mathbb {F}_r)$ given by scaling the length function. Specifically, $a \cdot (G,\rho ,\ell ) = (G,\rho ,a\cdot \ell )$ . The quotient of $\mathcal {X}(\mathbb {F}_r)$ is denoted $\mathbb {P}\mathcal {X}(\mathbb {F}_r)$ .

There are many continuous sections of the quotient map $\mathcal {X}(\mathbb {F}_r) \to \mathbb {P}\mathcal {X}(\mathbb {F}_r)$ . An often used choice uses the notion of the volume of a length function, $\operatorname {\mathrm {vol}}(\ell )$ , that we define now. For a length function $\ell \in \mathcal {M}(G)$ , we define the volume of $\ell $ by $\operatorname {\mathrm {vol}}(\ell ) = \sum _{e \in E_+}\ell (e)$ . There is a section $\mathcal {V} \colon \thinspace \mathbb {P}\mathcal {X}(\mathbb {F}_r) \to \mathcal {X}(\mathbb {F}_r)$ defined by

$$ \begin{align*} \mathcal{V}([ [(G,\rho,\ell)]]) = \bigg[\bigg(G,\rho,\frac{1}{\operatorname{\mathrm{vol}}(\ell)}\ell\bigg)\bigg]. \end{align*} $$

We denote the image of this section by $CV(\mathbb {F}_r)$ ; it is known as the Culler–Vogtmann outer space. Further, given a marked graph $\rho \colon \thinspace \mathcal {R}_r \to G$ , we set $CV(G,\rho ) = \mathcal {X}(G,\rho ) \cap CV(\mathbb {F}_r)$ . This set is homeomorphic to an open simplex of dimension $\lvert {E_+} \rvert - 1$ .

Example 2.1. There are three graphs in $\mathcal {G}_2$ : the $2$ -rose $\mathcal {R}_2$ , the theta graph $\Theta _2$ and the barbell graph $\mathcal {B}_2$ ; see Figure 2. Figure 3 shows a portion of $CV(\mathbb {F}_2)$ and how these simplices piece together. The homotopy equivalences used for Figure 3 are as follows:

$$ \begin{align*} \rho_\Theta \colon\thinspace \mathcal{R}_2 \to \Theta_2 & : e_1 \mapsto e_1\bar{e}_3, \, e_2 \mapsto e_2\bar{e}_3; \\ \rho_{\mathcal{B}} \colon\thinspace \mathcal{R}_2 \to \mathcal{B}_2 & : e_1 \mapsto e_1, \, e_2 \mapsto e_3e_2\bar{e}_3; \\ a \colon\thinspace \mathcal{R}_2 \to \mathcal{R}_2 & : e_1 \mapsto e_1, \, e_2 \mapsto \bar{e}_2; \\ b \colon\thinspace \mathcal{R}_2 \to \mathcal{R}_2 & : e_1 \mapsto \bar{e}_2, \, e_2 \mapsto e_1\bar{e}_2. \end{align*} $$

Notice that $\rho _\Theta $ is the homotopy inverse to the map $\Theta _2 \to \mathcal {R}_2$ that collapses the edge $e_3$ . Likewise $\rho _\Theta \circ b$ is the homotopy inverse to the collapse of $e_2$ and $\rho _\Theta \circ b^2$ is the homotopy inverse to the collapse of $e_1$ . Similarly, $\rho _{\mathcal {B}}$ is the homotopy inverse to the map $\mathcal {B}_2 \to \mathcal {R}_2$ that collapses $e_3$ .

Figure 2

The three homeomorphism types of graphs in $\mathcal {G}_2$ .

Figure 3

A portion of the Culler–Vogtmann outer space $CV(\mathbb {F}_2)$ .

One of the goals of this paper is to investigate a different continuous section of the quotient map. This uses the notion of entropy, $\mathfrak {h}_G(\ell )$ , defined in §3.1. Using this notion, there is a section $\mathcal {H} \colon \thinspace \mathbb {P}\mathcal {X}(\mathbb {F}_r) \to \mathcal {X}(\mathbb {F}_r)$ defined by

$$ \begin{align*} \mathcal{H}([ [(G,\rho,\ell)]]) = [(G,\rho,\mathfrak{h}_G(\ell)\ell)]. \end{align*} $$

We will denote the image of this section by $\mathcal {X}^1(\mathbb {F}_r)$ .

3.1 Entropy

Fix a finite connected graph $G = (V,E,o,\tau ,\bar {\phantom {e}})$ . An edge path $(e_1,\ldots ,e_n)$ in a graph G is reduced if $e_{i} \neq \bar {e}_{i+1}$ for $i = 1,\ldots ,n-1$ . A reduced edge path $(e_1,\ldots ,e_n)$ is a based circuit if $\tau (e_{n}) = o(e_{1})$ and $e_{n} \neq \bar {e}_{1}$ . The set of all based circuits in G is denoted by $\mathcal {C}(G)$ . For a length function $\ell \in \mathcal {M}(G)$ and a real number $t \geq 0$ , define

$$ \begin{gather*} \mathcal{C}_{G,\ell}(t) = \{\gamma \in \mathcal{C}(G) \mid \ell(\gamma) \leq t\}. \end{gather*} $$

Definition 3.1. The entropy of a length function $\ell \in \mathcal {M}(G)$ is

$$ \begin{align*} \mathfrak{h}_G(\ell) = \lim_{t \to \infty} \frac{1}{t} \log \lvert {\mathcal{C}_{G,\ell}(t)} \rvert. \end{align*} $$

Remark 3.2. We defined entropy as the growth rate of the number of reduced based circuits. In the literature, there exist many equivalent definitions of entropy. In particular, one can count the growth of reduced edge paths in G starting at a particular vertex and the adjective ‘based’ can be removed from the count of circuits [Reference Kapovich and Rivin24, Proposition 2.3]. This shows that $\mathfrak {h}_G(\ell )$ equals the volume entropy of $(G,\ell )$ . The volume entropy is defined as the exponential growth rate of the volume of balls in $({\widetilde G}, g_\ell )$ , where ${\widetilde G}$ is the universal cover of G and $g_\ell $ is the piecewise Riemannian metric obtained by pulling back the length function $\ell $ . That is,

$$ \begin{align*} \mathfrak{h}_G(\ell) = \lim_{t \to \infty} \frac{1}{t}\log \operatorname{\mathrm{vol}}_{g_\ell} B(x,t) \end{align*} $$

where $B(x,t)$ is the ball of radius t centered at $x \in {\widetilde G}$ , which is an arbitrary basepoint.

Example 3.3. The number of reduced edge paths in $\mathcal {R}_{r}$ with $\mathbf {1}$ -length equal to n is exactly $2r(2r-1)^{n-1}$ . Thus for any vertex $v \in \widetilde {\mathcal {R}}_r$ we have

$$ \begin{align*} \operatorname{\mathrm{vol}}_{g_{\mathbf{1}}} B(v,n) = \frac{r}{r-1}((2r-1)^{n} - 1). \end{align*} $$

Hence $\mathfrak {h}_{\mathcal {R}_r}(\mathbf {1}) = \log (2r-1)$ .

The next lemma shows that entropy is homogeneous of degree $-1$ and thus any length function $\ell \in \mathcal {M}(G)$ can be scaled to have unit entropy. Specifically, $\mathfrak {h}_G(a \cdot \ell ) = 1$ if and only if $a = \mathfrak {h}_G(\ell )$ .

Lemma 3.4. Let G be a finite connected graph and fix $\ell \in \mathcal {M}(G)$ . If $a \in \mathbb {R}_{>0}$ , then

$$ \begin{align*}\mathfrak{h}_G(a \cdot \ell) = \frac{1}{a}\mathfrak{h}_G(\ell).\end{align*} $$

Proof. We reparametrize the limit defining entropy by setting $s = a t$ . Then

$$ \begin{align*} \mathfrak{h}_G(\ell) & = \lim_{t \to \infty} \frac{1}{t} \log \lvert {\{\gamma \in \mathcal{C}(G) \mid \ell(\gamma) \leq t\}} \rvert \\ & = \lim_{s \to \infty} \frac{a}{s} \log \lvert {\{\gamma \in \mathcal{C}(G) \mid a \cdot \ell(\gamma) \leq s\}} \rvert = a\mathfrak{h}_G(a \cdot \ell).\\[-40pt] \end{align*} $$

Entropy defines an $\operatorname {\mathrm {Out}}(\mathbb {F}_r)$ -invariant function on $\mathcal {X}(\mathbb {F}_r)$ by $\mathfrak {h}([(G,\rho ,\ell )]) = \mathfrak {h}_G(\ell )$ . This function was investigated by Kapovich and Nagnibeda, who showed the following theorem.

In particular, the map $\mathcal {H} \colon \thinspace \mathbb {P}\mathcal {X}(\mathbb {F}_r) \to \mathcal {X}(\mathbb {F}_r)$ defined by normalizing to have unit entropy,

$$ \begin{align*} \mathcal{H}([ [(G,\rho,\ell)]]) = [(G,\rho,\mathfrak{h}_G(\ell)\ell)], \end{align*} $$

is a section. Hence the image $\mathcal {X}^1(\mathbb {F}_r) = \{[(G,\rho ,\ell )] \in \mathcal {X}(\mathbb {F}_r) \mid \mathfrak {h}_G(\ell ) = 1 \}$ is homeomorphic to $CV(\mathbb {F}_r)$ .

3.2 Pressure

Fix a finite connected graph $G = (V,E,o,\tau ,\bar {\phantom {e}})$ . We assume throughout this subsection and the next that $\chi (G) < 0$ (where $\chi (G) = \lvert {V} \rvert - \tfrac 12\lvert {E} \rvert $ is the Euler characteristic of G—note the $\tfrac 12$ factor is present as E includes edges with both orientations) and that G has no vertices with valence equal to $1$ or $2$ .

Define $A_{G} \in \operatorname {\mathrm {Mat}}_{\lvert {E} \rvert }(\mathbb {R})$ by

(3.1) $$ \begin{align} A_{G}(e,e') = \begin{cases} 1 & \mbox{if } \tau(e) = o(e') \mbox{ and } \bar{e} \neq e', \\ 0 & \mbox{otherwise. } \end{cases} \end{align} $$

It follows that the entry $A_{G}^{n}(e,e')$ is the number of reduced edge paths of the form $(e_{1},\ldots , e_{n})$ where $e_1 = e$ , $\tau (e_{n}) = o(e')$ and $\bar {e}_n \neq e'$ . In particular, $\operatorname {\mathrm {tr}}(A_{G}^{n})$ is the number of based edge circuits with $\mathbf {1}$ -length equal to n. Denoting the spectral radius of a matrix by , we get, from the definition of entropy, that

(3.2) $$ \begin{align} \mathfrak{h}_G(\mathbf{1}) = \log(\operatorname{\mathrm{spec}}(A_{G})). \end{align} $$

We remark that the above assumptions on G ensure that $A_G$ is irreducible.

In order to get a matrix that incorporates the metric and is related to entropy, we scale the rows of $A_G$ as follows: given a function $f \colon \thinspace E \to \mathbb {R}$ , we define $A_{G,f} \in \operatorname {\mathrm {Mat}}_{\lvert {E} \rvert }(\mathbb {R})$ by

(3.3) $$ \begin{align} A_{G,f}(e,e') = A_{G}(e,e')\exp(-f(e)). \end{align} $$

As for $A_{G}$ , it follows that $A_{G,f}^{n}(e,e')$ is the sum of $\exp (-f(\gamma ))$ over all edge paths of the form $\gamma = (e_{1},\ldots , e_{n})$ where $e_1 = e$ , $\tau (e_{n}) = o(e')$ and $\bar {e}_n \neq e'$ .

By equation (3.2) we have that $\mathfrak {P}_G(\mathbf {0}) = \mathfrak {h}_G(\mathbf {1})$ as $A_{G,-\mathbf {0}} = A_G$ , where $\mathbf {0}$ is the zero function.

The connection between entropy and pressure is given by the following theorem.

Let $\mathcal {M}^1(G) = \{ \ell \in \mathcal {M}(G) \mid \mathfrak {h}_G(\ell ) = 1 \}$ . By the first item above, we have that alternatively $\mathcal {M}^{1}(G) = \{ \ell \in \mathcal {M}(G) \mid \mathfrak {P}_G(-\ell ) = 0 \}$ . To see that $\mathcal {M}^1(G)$ is a codimension- $1$ submanifold of $\mathbb {R}^{\lvert {E_{+}} \rvert }$ we need to argue that $1$ is a regular value of $\mathfrak {h}_G$ . This follows from the following lemma. We denote the standard Euclidean inner product on $\mathbb {R}^n$ by .

Proof. This follows from the homogeneity of the entropy function (Lemma 3.4). Indeed,

$$ \begin{align*} \langle \ell,\nabla \mathfrak{h}_G(\ell) \rangle &= \lim_{s \to 0} \frac{\mathfrak{h}_G(\ell + s\ell) - \mathfrak{h}_G(\ell)}{s} = \lim_{s \to 0} \frac{\mathfrak{h}_G((1+s)\ell) - \mathfrak{h}_G(\ell)}{s} \\ &= \lim_{s \to 0} \frac{\frac{1}{1+s}\mathfrak{h}_G(\ell) - \mathfrak{h}_G(\ell)}{s} = \mathfrak{h}_G(\ell) \lim_{s \to 0} \frac{-s}{s(s+1)} = -\mathfrak{h}_G(\ell).\\[-46pt] \end{align*} $$

We record the following properties of the partial derivatives and the gradient of the pressure function. Given a function $f \colon \thinspace \mathbb {R}^{\lvert {E_+} \rvert } \to \mathbb {R}$ and an edge $e \in E_+$ , we denote the partial derivative of f with respect to the eth coordinate by $\partial _{e} f$ . Let denote the usual $L^{1}$ -norm on vectors in $\mathbb {R}^{n}$ .

3.3 Thermodynamic metrics

Fix a finite connected graph $G = (V,E,o,\tau ,\bar {\phantom {e}})$ . As in the previous section, we assume that $\chi (G) < 0$ and that G has no vertices with valence equal to $1$ or $2$ . The tangent space $T_{\ell }\mathcal {M}^{1}(G)$ at the length function $\ell \in \mathcal {M}^{1}(G)$ is the space of vectors $\mathbf {v} \in \mathbb {R}^{\lvert {E_{+}} \rvert }$ such that $\langle \mathbf {v},\nabla \mathfrak {h}_G(\ell ) \rangle = 0$ . The tangent bundle $T\mathcal {M}^{1}(G)$ is the subspace of $\mathcal {M}^1(G) \times \mathbb {R}^{\lvert {E_{+}} \rvert }$ consisting of pairs $(\ell ,\mathbf {v})$ where $\mathbf {v} \in T_{\ell }\mathcal {M}^{1}(G)$ .

We now define two Riemannian metrics on $\mathcal {M}^{1}(G)$ . We denote the Hessian (that is, the matrix of second derivatives) of a smooth function $f \colon \thinspace \mathbb {R}^{n} \to \mathbb {R}$ by $\mathbf {H}[f(x)]$ .

Definition 3.10. Given a length function $\ell \in \mathcal {M}^{1}(G)$ and tangent vectors $\mathbf {v}_{1},\mathbf {v}_{2} \in T_{\ell } \mathcal {M}^{1}(G)$ we define the entropy metric by

$$ \begin{align*} \langle \mathbf{v}_{1},\mathbf{v}_{2} \rangle_{\mathfrak{h},G} &= \langle \mathbf{v}_{1},\mathbf{H}[\mathfrak{h}_G(\ell)]\mathbf{v}_{2} \rangle, \end{align*} $$

and the pressure metric by

$$ \begin{align*} \langle \mathbf{v}_{1},\mathbf{v}_{2} \rangle_{\mathfrak{P},G} &= \langle \mathbf{v}_{1},\mathbf{H}[\mathfrak{P}_G(-\ell)]\mathbf{v}_{2} \rangle. \end{align*} $$

The associated norms on the tangent bundle $T\mathcal {M}^{1}(G)$ are denoted by

$$ \begin{align*} \| {(\ell,\mathbf{v})} \|_{\mathfrak{h},G}^{2} &= \langle \mathbf{v},\mathbf{H}[\mathfrak{h}_G(\ell)]\mathbf{v} \rangle \quad \mbox{and} \quad \| {(\ell,\mathbf{v})} \|_{\mathfrak{P},G}^{2} = \langle \mathbf{v},\mathbf{H}[\mathfrak{P}_G(-\ell)]\mathbf{v} \rangle. \end{align*} $$

By Theorem 3.7(3) we have that is positive definite. Positive definiteness of on $T_\ell \mathcal {M}^1(G)$ has been noted by others, but also follows from the positive definiteness of by Proposition 3.12.

The following proposition shows that these metrics lie in the same conformal class and that they can be calculated using the second derivative along a path. This feature is essential particularly for the material in §10.

Proposition 3.12. Let G be a finite connected graph. If $\ell _{t} \colon \thinspace (-1,1) \to \mathcal {M}^{1}(G)$ is a smooth path, then

$$ \begin{align*} \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{h},G}^{2} = -\langle \ddot\ell_{t},\nabla \mathfrak{h}_G(\ell_{t}) \rangle \quad \mbox{and} \quad \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{P},G}^{2} = \langle \ddot\ell_{t},\nabla \mathfrak{P}_G(-\ell_{t}) \rangle. \end{align*} $$

Additionally, given a length function $\ell \in \mathcal {M}^{1}(G)$ and tangent vectors $\mathbf {v}_{1},\mathbf {v}_{2} \in T_{\ell } \mathcal {M}^{1}(G)$ , we have

$$ \begin{align*} \langle \mathbf{v}_{1},\mathbf{v}_{2} \rangle_{\mathfrak{h},G} = \frac{\langle \mathbf{v}_{1},\mathbf{v}_{2} \rangle_{\mathfrak{P},G}}{\langle \ell,\nabla \mathfrak{P}_G(-\ell) \rangle}. \end{align*} $$

Proof. Differentiating the equation $\mathfrak {P}_G(-\ell _{t}) = 0$ with respect to t, we have $\langle \dot \ell _{t}, \nabla \mathfrak {P}_G(-\ell _{t}) \rangle = 0$ . Differentiating again, we find that

$$ \begin{align*} \langle \ddot \ell_{t}, \nabla\mathfrak{P}_G(-\ell_{t}) \rangle - \langle \dot \ell_{t},\mathbf{H} [\mathfrak{P}_G(-\ell_{t})]\dot\ell_{t} \rangle = 0. \end{align*} $$

Hence $\| {(\ell _{t},\dot \ell _{t})} \|_{\mathfrak {P}}^{2} = \langle \dot \ell _{t},\mathbf {H} [\mathfrak {P}(-\ell _{t})]\dot \ell _{t} \rangle = \langle \ddot \ell _{t},\nabla \mathfrak {P}(-\ell _{t}) \rangle $ as claimed.

The proof of the analogous statement for the entropy norm is similar, observing that $\mathfrak {h}_G(\ell _t) = 1$ .

By Lemma 3.8, we have that $\langle \ell ,\nabla \mathfrak {h}_G(\ell ) \rangle = -1$ for any $\ell \in \mathcal {M}^{1}(G)$ . Further, we have that $\nabla \mathfrak {P}_G(-\ell )$ is non-zero by Lemma 3.9 and hence is parallel to $\nabla \mathfrak {h}_G(\ell )$ by Theorem 3.7(1). Hence we find that

$$ \begin{align*} \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{h},G}^{2} \kern1.2pt{=}\kern1.2pt{-}\kern1.2pt\langle \ddot\ell_{t},\kern-1pt\nabla\mathfrak{h}_G(\ell_{t}) \rangle \kern1.2pt{=}\kern1.2pt \frac{\langle \ddot\ell_{t},\kern-1pt\nabla\mathfrak{h}_G(\ell_{t}) \rangle}{\langle \ell_{t},\kern-1pt\nabla \mathfrak{h}_G(\ell_{t}) \rangle} \kern1.2pt{=}\kern1.2pt \frac{\langle \ddot\ell_{t},\kern-1pt\nabla\mathfrak{P}_G(-\ell_{t}) \rangle}{\langle \ell_{t},\kern-1pt\nabla \mathfrak{P}_G(-\ell_{t}) \rangle} \kern1.2pt{=}\kern1.2pt \frac{\| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{P},G}^{2}}{\langle \ell_{t},\kern-1pt\nabla \mathfrak{P}_G(-\ell_{t}) \rangle}\kern-1pt. \end{align*} $$

By polarization, the norm determines the inner product and so the claim follows.

Positive definiteness of the Hessians follows from strict convexity of $\mathfrak {h}_G$ and $\mathfrak {P}_G$ on $\mathcal {M}(G)$ and $\mathbb {R}^{\lvert {E_{+}} \rvert }$ , respectively (Theorem 3.7).

Using these norms, we can define the entropy or pressure length of a piecewise smooth path $\ell _{t} \colon \thinspace [t_0,t_1] \to \mathcal {M}^{1}(G)$ by

$$ \begin{align*} \mathcal{L}_{\mathfrak{h},G}(\ell_{t}|[t_0,t_1])&=\! \int_{t_0}^{t_1} \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{h},G} \, dt,\\ \mathcal{L}_{\mathfrak{P},G}(\ell_{t}|[t_0,t_1])&=\! \int_{t_0}^{t_1} \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{P},G} \, dt. \end{align*} $$

These induce the entropy and pressure distance functions on $\mathcal {M}^1(G)$ by

$$ \begin{align*} d_{\mathfrak{h},G}(x,y) &= \inf\{ \mathcal{L}_{\mathfrak{h},G}(\ell_{t}|[0,1]) \mid \ell_{t} \colon\thinspace [0,1] \to \mathcal{M}^{1}(G), \, \ell_{0} = x, \, \ell_{1} = y \},\\ d_{\mathfrak{P},G}(x,y) &= \inf\{ \mathcal{L}_{\mathfrak{P},G}(\ell_{t}|[0,1]) \mid \ell_{t} \colon\thinspace [0,1] \to \mathcal{M}^{1}(G), \, \ell_{0} = x, \, \ell_{1} = y \}. \end{align*} $$

Given a marked graph $(G,\rho )$ , we set $\mathcal {X}^1(G,\rho ) = \mathcal {X}(G,\rho ) \cap \mathcal {X}^1(\mathbb {F}_r)$ . Using the natural bijection $\mathcal {X}^1(G,\rho ) \leftrightarrow \mathcal {M}^1(G)$ , we get metrics and distance functions on $\mathcal {X}^1(G,\rho )$ that we denote using the same notation as above.

Next, we explain how these fit together to get distance functions on $\mathcal {X}^1(\mathbb {F}_r)$ . Suppose $\alpha _t \colon \thinspace [0,1] \to \mathcal {X}^1(\mathbb {F}_r)$ is a piecewise smooth path and that there is a partition $t_1 = 0 < t_2 < \cdots < t_{n+1} = 1$ and marked graphs $(G_k,\rho _k)$ for $k = 1,\ldots ,n$ such that $\alpha _t \in \mathcal {X}^1(G_k,\rho _k)$ for $t \in (t_k,t_{k+1})$ . We set

$$ \begin{align*} \mathcal{L}_{\mathfrak{h}}(\alpha_t) & = \sum_{k=1}^n \mathcal{L}_{\mathfrak{h},G_k}(\alpha_t | (t_k,t_{k+1})) \quad \mbox{and} \quad \mathcal{L}_{\mathfrak{P}}(\alpha_t) = \sum_{k=1}^n \mathcal{L}_{\mathfrak{P},G_k}((\alpha_t | (t_k,t_{k+1})). \end{align*} $$

These define distance functions on $\mathcal {X}^1(\mathbb {F}_r)$ as usual—by taking the infimum of the lengths of paths—that we denote by $d_{\mathfrak {h}}$ and $d_{\mathfrak {P}}$ . It is obvious that the proposed distance functions are symmetric and satisfy the triangle inequality; positive definiteness of the Hessians implies non-degeneracy. However, it is not obvious that the distances they define are finite: a priori it may be possible that the length of a path that collapses an edge is infinite. This will be addressed in §5.

4.1 Determinant equation

Fix a finite connected graph $G = (V,E,o,\tau ,\bar {\phantom {e}})$ . We always assume that $\chi (G) < 0$ and G has no vertices of valence $1$ or $2$ .

Let $D_{G}$ be the directed graph with adjacency matrix $A_{G}$ . Thus the vertex set for $D_G$ is the set E (recall our notation $E = E_+ \cup \overline {E}_+$ ) and there is an edge from e to $e'$ if $A_G(e,e') = 1$ , that is, if $\tau (e) = o(e')$ and $\bar {e} \neq e'$ . The cycle complex of $D_{G}$ , denoted by $C_{G}$ , is the abstract simplicial complex with an n-simplex for each collection $\Delta = \{ z_{1},\ldots ,z_{n+1}\}$ of pairwise disjoint simple cycles, that is, embedded loops, in $D_{G}$ .

Example 4.1. Suppose that G is the barbell graph as shown below:

Order the edges of G by a, $\bar {a}$ , b, $\bar {b}$ , c, $\bar {c}$ . The matrix $A_{G}$ and directed graph $D_{G}$ are as presented below:

There are eight simple cycles in $D_G$ : $\gamma _{a} = (a)$ , $\gamma _{\bar {a}} = (\bar {a})$ , $\gamma _{b} = (b)$ , $\gamma _{\bar {b}} = (\bar {b})$ , $\gamma _{ab} = (a,c,b,\bar {c})$ , $\gamma _{a\bar {b}} = (a,c,\bar {b},\bar {c})$ , $\gamma _{\bar {a}b} = (\bar {a},c,b,\bar {c})$ and $\gamma _{\bar {a}\bar {b}} = (\bar {a},c,\bar {b},\bar {c})$ . The cycle complex $C_{G}$ is the flag complex whose 1-skeleton is shown in Figure 4.

Figure 4

The 1-skeleton of the cycle complex $C_G$ in Example 4.1.

Given a function $f \colon \thinspace E \to \mathbb {R}$ (in particular, a length function) and a simple cycle z in $D_G$ , we set $f(z) = \sum _{i=1}^n f(e_i)$ where $e_1,\ldots ,e_n$ are the vertices in $D_G$ traversed by z (each corresponding to an oriented edge in G). Likewise, for a simplex $\Delta = \{ z_{1},\ldots ,z_{n}\}$ in $C_G$ we set $f(\Delta ) = \sum _{k=1}^{n} f(z_{k})$ . We consider the empty set as a simplex and define $f(\emptyset ) = 0$ for any function $f \colon \thinspace E \to \mathbb {R}$ . Lastly, for a simplex $\Delta = \{z_1,\ldots ,z_n\}$ we set $\lvert {\Delta } \rvert = n$ .

Recall that given a length function $\ell \in \mathcal {M}(G)$ , the matrix $A_{G,\ell }$ is defined by $A_{G,\ell }(e,e') = \exp (-\ell (e))A_G(e,e')$ . We consider the function $F_{G} \colon \thinspace \mathcal {M}(G) \to \mathbb {R}$ given by

(4.1) $$ \begin{align} F_{G}(\ell) = \det(I - A_{G,\ell}). \end{align} $$

This function can be expressed using the cycle complex $C_G$ as follows.

Theorem 4.2. Let G be a finite connected graph and fix $\ell \in \mathcal {M}(G)$ . Then

$$ \begin{align*} F_{G}(\ell) = \sum_{\Delta \in C_{G}} (-1)^{\lvert {\Delta} \rvert} \exp(-\ell(\Delta)). \end{align*} $$

Example 4.3. We apply Theorem 4.2 to the case when G is the barbell graph as in Example 4.1. Using the change of variables $x = \exp (-\ell (a))$ , $y = \exp (-\ell (b))$ and $z = \exp (-\ell (c))$ , we find that

$$ \begin{align*} F_{G}(\ell) &= \, 1 - (2x + 2y + 4xyz^{2}) + (x^{2} + y^{2} + 4xy + 4x^{2}yz^{2} + 4xy^{2}z^{2} ) \\ &\quad -(2x^{2}y + 2xy^{2} + 4x^{2}y^{2}z^{2}) + x^{2}y^{2}. \end{align*} $$

The following statements show how the function $F_{G}$ is related to $\mathcal {M}^{1}(G)$ .

Proof. Since $\ell $ lies in $\mathcal {M}^{1}(G)$ , Theorem 3.7(1) and the definition of pressure imply that $\operatorname {\mathrm {spec}}(A_{G,\ell }) = 1$ . Above we remarked that the assumptions on G imply that $A_G$ is irreducible; hence so is $A_{G,\ell }$ . By the Perron–Frobenius theorem the spectral radius of $A_{G,\ell }$ is realized by a positive, real, simple eigenvalue; (1) follows.

Now consider the function $p \colon \thinspace \mathbb {R} \to \mathbb {R}$ defined by $p(t) = \det (I - tA_{G,\ell })$ . Let $1 = \lambda _{1}, \ldots , \lambda _{\lvert {E} \rvert }$ be the roots of the characteristic polynomial of $A_{G,\ell }$ . Then we can write

$$ \begin{align*} p(t) = (1-t)\prod_{i=2}^{\lvert {E} \rvert} (1-t\lambda_{i}). \end{align*} $$

Therefore, taking the derivative, we find that

$$ \begin{align*} p'(1) = -\prod_{i=2}^{\lvert {E} \rvert} (1-\lambda_{i}).\end{align*} $$

For $i = 2,\ldots , \lvert {E} \rvert $ we have that $|\lambda _i| \leq 1$ and $\lambda _i \neq 1$ . Combining these observations with the fact that complex eigenvalues come in conjugate pairs, it follows that $p'(1) < 0$ . Observe that $F_{G}(\ell + s \cdot \mathbf {1}) = \det (I - \exp (-s)A_{G,\ell }) = p(\exp (-s))$ . Therefore we have that $\langle \mathbf {1},\nabla F_{G}(\ell ) \rangle = -p'(1)\exp (0)> 0$ , giving (2).

We claim that $\nabla F_G(\ell )$ is parallel to $\nabla \mathfrak {P}_G(\ell )$ for $\ell \in \mathcal {M}^1(G)$ . Indeed, this follows as (1), (2) and Theorem 3.7 imply that $\mathfrak {P}_G^{-1}(0) = \mathcal {M}^1(G)$ is a connected component of $F_G^{-1}(0)$ . By (2) and Lemma 3.9(2) the gradients of $F_G$ and $\mathfrak {P}_G$ are non-zero for this subset. As gradients are always orthogonal to level sets, the claim follows. Hence by Lemma 3.9(1), we have that either $\partial _{e} F_G(\ell )> 0$ or $\partial _{e} F_G(\ell ) < 0$ for all $e \in E_+$ and $\ell \in \mathcal {M}^1(G)$ . Since $\langle \mathbf {1},\nabla F_G(\ell ) \rangle>0$ , we must have the former, whence (3).

As a remarked in the proof of Lemma 4.4, we have the following corollary.

Using these observations, we can compute the entropy and pressure norms using the function $F_{G}$ .

Proposition 4.6. If $\ell _{t} \colon \thinspace (-1,1) \to \mathcal {M}^{1}(G)$ is a smooth path, then

$$ \begin{align*} \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{h},G}^{2} &= \frac{-\langle \dot\ell_{t},\mathbf{H} [F_{G}(\ell_{t})]\dot\ell_{t} \rangle}{\langle \ell_{t},\nabla F_{G}(\ell_{t}) \rangle}= \frac{\langle \ddot\ell_{t},\nabla F_{G}(\ell_{t}) \rangle}{\langle \ell_{t},\nabla F_{G}(\ell_{t}) \rangle}, \\ \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{P},G}^{2} &= \frac{-\langle \dot\ell_{t},\mathbf{H} [F_{G}(\ell_{t})]\dot\ell_{t} \rangle}{\| {\nabla F_{G}(\ell_{t})} \|_{1}}= \frac{\langle \ddot\ell_{t},\nabla F_{G}(\ell_{t}) \rangle}{\| {\nabla F_{G}(\ell_{t})} \|_{1}}. \end{align*} $$

Using Theorem 4.2, we can compute the partial derivatives of $F_G$ . We find for any edges $e,e' \in E_+$ that

(4.2) $$ \begin{align} \hspace{-8pt}\partial_{e} F_{G}(\ell) &= -\sum_{\Delta \in C_G} (-1)^{\lvert {\Delta} \rvert}\Delta(e)\exp(-\ell(\Delta)), \end{align} $$

(4.3) $$ \begin{align} \partial_{e e'} F_{G}(\ell) &= \sum_{\Delta \in C_G} (-1)^{\lvert {\Delta} \rvert}\Delta(e)\Delta(e')\exp(-\ell(\Delta)), \end{align} $$

where $\Delta (e) \in \{0,1,2\}$ denotes the cardinality of the intersection $\{e,\bar {e}\} \cap \Delta $ . Using this notation, we remark that $\ell (\Delta ) = \sum _{e \in E_+} \Delta (e)\ell (e)$ for a length function $\ell \in \mathcal {M}(G)$ and simplex $\Delta \in C_G$ . Given a vector $\mathbf {v} \in \mathbb {R}^{\lvert {E_+} \rvert }$ and a simplex $\Delta \in C_G$ , we set $\mathbf {v}(\Delta ) = \sum _{e \in E_+} \Delta (e)\mathbf {v}(e)$ . Using these expressions, we can rewrite the dot products appearing in the formulas for the metrics in Proposition 4.6 as sums over simplices in $C_{G}$ rather than over the edges of G as follows.

Lemma 4.7. For $\ell \in \mathcal {M}(G)$ , we have

$$ \begin{align*} \langle \ell, \nabla F_{G}(\ell) \rangle = -\sum_{\Delta \in C_{G}} (-1)^{\lvert {\Delta} \rvert}\ell(\Delta)\exp(-\ell(\Delta)). \end{align*} $$

Proof. We compute:

$$ \begin{align*} \langle \ell, \nabla F_{G}(\ell) \rangle & = \sum_{e \in E_{+}} \ell(e) \bigg(-\sum_{\Delta \in {C_{G}}}(-1)^{\lvert {\Delta} \rvert}\Delta(e)\exp(-\ell(\Delta))\bigg) \\ & = -\sum_{\Delta \in {C_{G}}} (-1)^{\lvert {\Delta} \rvert}\exp(-\ell(\Delta)) \bigg(\sum_{e \in E_{+}} \ell(e)\Delta(e)\bigg) \\ & = -\sum_{\Delta \in C_{G}} (-1)^{\lvert {\Delta} \rvert}\ell(\Delta)\exp(-\ell(\Delta)).\\[-47pt] \end{align*} $$

Lemma 4.8. Let G be a finite connected graph. If $(\ell , \mathbf {v}) \in T\mathcal {M}^{1}(G)$ , then

$$ \begin{align*} \langle \mathbf{v}, \mathbf{H}[F_{G}(\ell)]\mathbf{v} \rangle = \sum_{\Delta \in C_{G}} (-1)^{\lvert {\Delta} \rvert}\mathbf{v}(\Delta)^{2}\exp(-\ell(\Delta)). \end{align*} $$

Proof. This is similar to Lemma 4.7. The eth component of $\mathbf {H}[F_{G}(\ell )]\mathbf {v}$ is

$$ \begin{align*}\sum_{e' \in E_{+}}\! \!\bigg(\! \sum_{\Delta \in C_{G}} (-1)^{\lvert {\Delta} \rvert} \Delta(e)\Delta(e')\exp(-\ell(\Delta))\bigg) \mathbf{v}(e') =\! \sum_{\Delta \in C_{G}}\! (-1)^{\lvert {\Delta} \rvert} \Delta(e)\mathbf{v}(\Delta)\exp(-\ell(\Delta)). \end{align*} $$

Hence

$$ \begin{align*} \langle \mathbf{v}, \mathbf{H}[\nabla F_{G}(\ell)]\mathbf{v} \rangle & = \sum_{e \in E_{+}} \mathbf{v}(e) \bigg(\sum_{\Delta \in {C_{G}}}(-1)^{\lvert {\Delta} \rvert}\Delta(e)\mathbf{v}(\Delta)\exp(-\ell(\Delta))\bigg) \\ &= \sum_{\Delta \in C_{G}} (-1)^{\lvert {\Delta} \rvert}\mathbf{v}(\Delta)^{2}\exp(-\ell(\Delta)).\\[-46pt] \end{align*} $$

By Lemma 4.4, if $\mathfrak {h}_G(\ell ) = 1$ then $F_{G}(\ell ) = 0$ . The next lemma gives a partial converse.

Proof. We begin by showing that if $\mathfrak {h}_G(\ell ) < 1$ , then $F_G(\ell ) \neq 0$ . To begin, we observe that if $\mathfrak {h}_G(\ell ) < 1$ , then $\mathfrak {P}_G(-\ell ) < 0$ . Indeed, let $\ell _t \colon \thinspace [0,1] \to \mathcal {M}(G)$ be the path defined by

$$ \begin{align*} \ell_t = (1-t)\ell + t\mathfrak{h}_G(\ell)\ell. \end{align*} $$

We have that $\langle \nabla \mathfrak {P}_G(-\ell _t), - \dot \ell _t \rangle> 0$ as each component of $\nabla \mathfrak {P}_G(-\ell _t)$ is positive by Lemma 3.9(1) and each component of $-\dot \ell _t$ is positive by construction. Notice that $\mathfrak {P}_G(-\ell _1) = 0$ by Theorem 3.7(1) since $\mathfrak {h}_G(\ell _1) = 1$ . Therefore, we find that

$$ \begin{align*} -\mathfrak{P}_G(-\ell_0) = \int_0^1 \langle \nabla \mathfrak{P}_G(-\ell_t),-\dot \ell_t \rangle \, dt> 0 \end{align*} $$

and hence $\mathfrak {P}_G(-\ell ) = \mathfrak {P}_G(-\ell _0)$ is negative as claimed. Therefore we have that $\operatorname {\mathrm {spec}} (A_{G,\ell }) < 1$ and, in particular, $1$ is not an eigenvalue of $A_{G,\ell }$ . Hence $F_G(\ell ) \neq 0$ . This completes the claim that $F_G(\ell ) \neq 0$ for any $\ell \in \mathcal {M}(G)$ where $\mathfrak {h}_G(\ell ) < 1$ .

We now complete the proof of the lemma. Suppose that $\mathfrak {h}_G(\ell ) < 1$ and consider the continuous function $p \colon \thinspace [0,\infty ) \to \mathbb {R}$ defined by $p(t) = F_G(\ell + t \cdot \mathbf {1})$ . As $\mathfrak {h}_G(\ell + t \cdot \mathbf {1}) < 1$ for all $t \in [0,\infty )$ , by the above claim we have that $p(t) \neq 0$ . Since ${p(t) \to 1}$ as $t \to \infty $ , we have that $p(t)> 0$ for all $t \in [0,\infty )$ . In particular, we have that $F_G(\ell ) = p(0)> 0$ .

4.2 A simplification

The function $F_{G}$ factors non-trivially in special cases as a result of certain aspects of the graph G. In such a case, we can replace $F_G$ with one of these factors and simplify the expressions for the entropy and pressure norm from Proposition 4.6.

For instance, one factorization occurs if e is a loop edge. When $\ell (e) = 0$ the vector $\mathbf {v} \in \mathbb {R}^{\lvert {E} \rvert }$ , where $\mathbf {v}(e) = 1$ , $\mathbf {v}(\bar {e}) = -1$ , and the rest of the entries are equal to 0, is an eigenvector of $A_{G,\ell }$ with eigenvalue 1. This means $1-\exp (-\ell (e))$ is a factor of $F_{G}$ .

Example 4.10. Using the notation from Example 4.3, we have that both $1-x$ and $1-y$ are factors of $F_{G}$ . Factoring, we have

$$ \begin{align*} F_{G}(\ell) = (1-x)(1-y)(1 - x - y + xy - 4xyz^{2}). \end{align*} $$

Another case where there is a factorization of $F_{G}$ is when the edge involution $e \leftrightarrow \bar {e}$ is a graph automorphism of G. There are only two types of graphs for which such an automorphism exists:

1. (1) the r-rose, $\mathcal {R}_{r}$ ;

2. (2) the graph $\Theta _{r}$ with vertices, v and w, and edges $e_1, \ldots ,e_{r+1}$ where $o(e_i) = v$ and $\tau (e_i) = w$ for $i = 1,\ldots ,r+1$ .

In this case ordering the edges in $E_+$ first and then ordering the edges in $E - E_+$ accordingly, we have that

$$ \begin{align*} A_G = \begin{bmatrix} B_G & B^{\prime}_G \\ B^{\prime}_G & B_G \end{bmatrix} \end{align*} $$

for two matrices $B_G,B^{\prime }_G \in \operatorname {\mathrm {Mat}}_{\lvert {E_+} \rvert }(\mathbb {R})$ . Thus

$$ \begin{align*} F_G(\ell) &= \det(I - A_{G,\ell}) = \det \begin{bmatrix} I - B_{G,\ell} & -B^{\prime}_{G,\ell} \\ -B^{\prime}_{G,\ell} & I - B_{G,\ell} \end{bmatrix} \\ & = \det(I - B_{G,\ell} - B^{\prime}_{G,\ell})\det(I - B_{G,\ell} + B^{\prime}_{G,\ell}). \end{align*} $$

Since each row of $B_G$ or $B^{\prime }_G$ corresponds to an edge $e \in E_+$ the notation $B_{G,\ell }$ and $B^{\prime }_{G,\ell }$ still makes sense.

For $\mathcal {R}_r$ , we have that $B_{\mathcal {R}_r}$ is the $r \times r$ matrix consisting of all $1$ s and $B^{\prime }_{\mathcal {R}_r} = B_{\mathcal {R}_r} - I$ . In this case

$$ \begin{align*} \det(I - B_{\mathcal{R}_r,\ell} + B^{\prime}_{\mathcal{R}_r,\ell}) = \prod_{e \in E_+}(1-\exp(-\ell(e))). \end{align*} $$

These are precisely the factors which were observed above for loop edges.

For $\Theta _{r}$ , we have that $B_{\Theta _r}$ is the $(r+1) \times (r+1)$ matrix consisting of all $0$ s and $B^{\prime }_{\Theta _r}$ is the $(r + 1) \times (r+1)$ matrix where all diagonal entries are $0$ and all non-diagonal entries are $1$ . In this case we have

$$ \begin{align*} F_{\Theta_{r}}(\ell) & = \det(I - B^{\prime}_{\Theta_r,\ell})\det(I + B^{\prime}_{\Theta_r,\ell}). \end{align*} $$

In general, we now construct a graph quotient $D_{G} \to \overline {D}_{G}$ that identifies certain edge pairs $\{e,\bar {e}\}$ resulting in a new matrix $\overline {A}_{G}$ , which selects the appropriate factor. In this new matrix, every row corresponds to either an edge $e \in E$ or an edge pair $\{ e,\bar {e}\}$ , and so we can still make sense of $\overline {A}_{G,f}$ for a function $f \colon \thinspace E_+ \to \mathbb {R}$ .

When G is $\mathcal {R}_r$ or $\Theta _{r}$ , we take $\overline {D}_{\Theta _{r}}$ to be the quotient of $D_{\Theta _{r}}$ by the orientation-reversing automorphism $e \mapsto \bar {e}$ . In this case $\overline {A}_{G,\ell } = B_{G,\ell } + B^{\prime }_{G,\ell }$ .

Otherwise, for each pair $\{e,\bar {e}\}$ that is a loop edge of G, we identify the vertices of $D_{G}$ corresponding to e and $\bar {e}$ , now denoted $e\bar {e}$ , keep the incoming edges and identify the outgoing edges that have the same terminal vertex. We call the resulting graph $\overline {D}_{G}$ . Algebraically, we add together the columns corresponding to e and $\bar {e}$ and delete one of the rows corresponding to e and $\bar {e}$ .

We define $\overline {F}_{G} \colon \thinspace \mathcal {M}(G) \to \mathbb {R}$ by

(4.4) $$ \begin{align} \overline{F}_{G}(\ell) = \det(I - \overline{A}_{G,\ell}). \end{align} $$

The formula in Theorem 4.2, the formulas for the partial derivatives in (4.2) and (4.3), and the inner products in Lemmas 4.7 and 4.8 hold for $\overline {F}_{G}$ using the complex $\overline {C}_G$ , which is the cycle complex of the directed graph $\overline {D}_G$ .

Example 4.11. For G equal to the barbell graph as in Example 4.1 we have $\overline {A}_{G}$ and $\overline {D}_{G}$ as shown below (columns of the matrix are ordered as $a\bar {a}$ , $b\bar {b}$ , c, $\bar {c}$ ):

The two directed edges from $\bar {c}$ to $a\bar {a}$ are identified with the set $\{a ,\bar {a}\}$ so that we think of the sequence $c,a$ or $c,\bar {a}$ as specifying which of the two edges between $\bar {c}$ and $a\bar {a}$ to traverse in a cycle. There are six simple cycles: $\gamma _{a\bar {a}} = (a\bar {a})$ , $\gamma _{b\bar {b}} = (b\bar {b})$ , $\gamma _{ab} = (a,c,b,\bar {c})$ , $\gamma _{a\bar {b}} = (a,c,\bar {b},\bar {c})$ , $\gamma _{\bar {a}b} = (\bar {a},c,b,\bar {c})$ and $\gamma _{\bar {a}\bar {b}} = (\bar {a},c,\bar {b},\bar {c})$ . The cycle complex $\overline {C}_{G}$ is shown below:

Using Theorem 4.2, we find (with $x = \exp (-\ell (a))$ , $y = \exp (-\ell (b))$ and $z = \exp (-\ell (c))$ ) that

$$ \begin{align*} \overline{F}_{G}(\ell) =& \, \sum_{\Delta \in \overline{C}_{G}} (-1)^{\lvert {\Delta} \rvert} \exp(-\ell(\Delta)) = 1 - (x + y + 4xyz^{2}) + xy. \end{align*} $$

The astute reader will notice the comparison with Example 4.10.

In particular, we have that $\mathfrak {P}_G(-\ell ) = \log \operatorname {\mathrm {spec}}(\overline {A}_{G,\ell })$ . As in Corollary 4.5, we have the following statement. This follows for the same reasons as in Lemma 4.4 as $\overline {F}_G(\ell ) = 0$ and $\nabla \overline {F}_G(\ell ) \neq 0$ for $\ell \in \mathcal {M}^1(G)$ .

We also observe that the formulas for the metrics in Proposition 4.6 hold for $\overline {F}_G$ .

Proposition 4.14. If $\ell _{t} \colon \thinspace (-1,1) \to \mathcal {M}^{1}(G)$ is a smooth path, then

$$ \begin{align*} \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{h},G}^{2} &= \frac{-\langle \dot\ell_{t},\mathbf{H} [\overline{F}_{G}(\ell_{t})]\dot\ell_{t} \rangle}{\langle \ell_{t},\nabla \overline{F}_{G}(\ell_{t}) \rangle}= \frac{\langle \ddot\ell_{t},\nabla \overline{F}_{G}(\ell_{t}) \rangle}{\langle \ell_{t},\nabla \overline{F}_{G}(\ell_{t}) \rangle}, \\ \| {(\ell_{t},\dot\ell_{t})} \|_{\mathfrak{P},G}^{2} &= \frac{-\langle \dot\ell_{t},\mathbf{H} [\overline{F}_{G}(\ell_{t})]\dot\ell_{t} \rangle}{\| {\nabla \overline{F}_{G}(\ell_{t})} \|_{1}}= \frac{\langle \ddot\ell_{t},\nabla \overline{F}_{G}(\ell_{t}) \rangle}{\| {\nabla \overline{F}_{G}(\ell_{t})} \|_{1}}. \end{align*} $$

6.1 The 2-rose

Denote the edges of $\mathcal {R}_2$ by $e_1$ and $e_2$ . To make the calculations in this subsection easier to read, given a length function $\ell \in \mathcal {M}(\mathcal {R}_2)$ , we set $a = \ell (e_1)$ , $b = \ell (e_2)$ and $m = \ell (e_1) + \ell (e_2)$ . Applying the definition of $\overline {F}_G$ from §4.2 to $\mathcal {R}_2$ , we find the formula

(6.1) $$ \begin{align} \overline{F}_{\mathcal{R}_2}(\ell) = 1 - \exp(-a) - \exp(-b) - 3\exp(-m). \end{align} $$

Lemma 6.1. Suppose $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^{1}(\mathcal {R}_{2})$ is a smooth path such that for all $t \in [0,1]$ we have $\dot m_t> 0$ . If $m_0 \geq 4$ , then

$$ \begin{align*} \mathcal{L}_{\mathfrak{h},\mathcal{R}_2}(\ell_t|[0,1]) \geq \sqrt{m_1} - \sqrt{m_0}. \end{align*} $$

Proof. Suppose that $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^1(\mathcal {R}_2)$ is a path where $\dot m_t> 0$ as in the statement of the lemma and assume that $m_0 \geq 4$ . We reparametrize the path $\ell _t$ so that $m_t = t$ .

Let $n_t = \min \{a_t,b_t\}$ . As $\overline {F}_{\mathcal {R}_2}(\ell _t) = 0$ , we have

$$ \begin{align*} 1 - 2\exp(-n_t) \leq 1 - \exp(-a_t) - \exp(-b_t) = 3\exp(-m_t) = 3\exp(-t). \end{align*} $$

In particular, $2\exp (-n_t) \geq 1 - 3\exp (-t) \geq 2\exp (-1)$ as $t \geq 4$ and so $n_t < 1$ for all t.

Setting $p_t = \max \{a_t,b_t\}$ , we find that $p_t = t - n_t \geq t-1$ . Therefore, as $t - 1 \geq 1$ for $t \geq 4$ , the edge that realizes the minimum of $\{a_t,b_t\}$ does not depend on t and hence we may assume that $b_t = n_t$ and that $a_t = p_t \geq t-1$ . This gives us that $\exp (-a_t) \leq \exp (-t + 1)$ . Hence, again as $\overline {F}_{\mathcal {R}_2}(\ell _t) = 0$ , we have

$$ \begin{align*} 1 - \exp(-b_t) = \exp(-a_t) + 3\exp(-m_t) \leq (\exp(1) + 3)\exp(-t) \ \leq 8\exp(-t). \end{align*} $$

This enables us to give an upper bound on the denominator in the expression for the entropy norm. Specifically, using the fact that $x\exp (-x) \leq 1 - \exp (-x)$ for $x \geq 0$ , we have

$$ \begin{align*} \langle \ell_t,\nabla \overline{F}_{\mathcal{R}_2}(\ell_t) \rangle & = a_t\exp(-a_t) + b_t\exp(-b_t) +3m_t\exp(-m_t) \\ & \leq t\exp(-t+1) + (1 - \exp(-b_t)) +3t\exp(-t) \\ & \leq 8t\exp(-t) + 8\exp(-t) \\ & \leq 12t\exp(-t). \end{align*} $$

In the final inequality we used that fact that $t \geq 4$ . The expression for $\langle \ell _t,\nabla \overline {F}_{\mathcal {R}_2}(\ell _t) \rangle $ can be computed either directly from (6.1) or via Lemma 4.7.

Next, we get an upper bound on the numerator in the expression for the entropy norm by just using $m_t$ . Specifically,

$$ \begin{align*} -\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\mathcal{R}_2}(\ell_t)]\dot \ell_t \rangle & = (\dot a_t)^2\exp(-a_t) + (\dot b_t)^2\exp(-b_t) + 3(\dot m_t)^2\exp(-m_t) \\ & \geq 3\exp(-t). \end{align*} $$

As above, the expression for $-\langle \dot \ell _t,\mathbf {H}[\overline {F}_{\mathcal {R}_2}(\ell _t)]\dot \ell _t \rangle $ can be computed either directly from (6.1) or via Lemma 4.8.

Hence we find that the entropy norm along this path is bounded below by

$$ \begin{align*} \| {(\ell_t,\dot \ell_t)} \|^2_{\mathfrak{h},\mathcal{R}_2} = \frac{-\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\mathcal{R}_2}(\ell_t)]\dot \ell_t \rangle}{\langle \ell_t,\nabla \overline{F}_{\mathcal{R}_2}(\ell_t) \rangle}\geq \frac{1}{4t}. \end{align*} $$

Therefore the length of this path in the entropy metric is at least

$$ \begin{align*} \int_{m_0}^{m_1} \sqrt{\frac{1}{4t}} \, dt &= \sqrt{m_1} - \sqrt{m_0}. \\[-42pt]\end{align*} $$

Using this lemma, we can get a lower bound on the distance between length functions in $\mathcal {M}^1(\mathcal {R}_2)$ in terms of the sum of the lengths of edges, so long as they are sufficiently large.

Proposition 6.2. Suppose $\ell $ and $\ell '$ are length functions in $\mathcal {M}^1(\mathcal {R}_2)$ where $m = \ell (e_1) + \ell (e_2)$ and $m' = \ell '(e_1) + \ell '(e_2)$ are at least $4$ . Then

$$ \begin{align*} d_{\mathfrak{h},\mathcal{R}_2}(\ell,\ell') \geq \sqrt{m'} - \sqrt{m}. \end{align*} $$

Proof. Let $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^1(\mathcal {R}_2)$ be a piecewise smooth path such that $\ell _0 = \ell $ and $\ell _1 = \ell '$ . Let $\delta \in [0,1]$ be the minimal value such that $m_t \geq 4$ for $t \in [\delta ,1]$ . In particular, ${m_\delta \leq m}$ .

By only considering the smooth subpaths of $\ell _t|[\delta ,1]$ for which $\dot m_t> 0$ , by Lemma 6.1, we find that

$$ \begin{align*} \mathcal{L}_{\mathfrak{h},\mathcal{R}_2}(\ell_t|[\delta,1]) \geq \sqrt{m'} - \sqrt{m_\delta} \geq \sqrt{m'} - \sqrt{m}. \end{align*} $$

This also provides a lower bound on $\mathcal {L}_{\mathfrak {h},\mathcal {R}_2}(\ell _t|[0,1])$ . Since the path was arbitrary this also is a lower bound on the distance between $\ell $ and $\ell '$ .

6.2 The barbell graph

Let $\mathcal {B}_2$ denote the graph with vertices v and w, and edges $e_1$ , $e_2$ and $e_3$ where $o(e_1) = \tau (e_1) = v$ , $o(e_2) = \tau (e_2) = w$ , and $o(e_3) = v$ and $\tau (e_3) = w$ . To make the calculations in this section easier to read, given a length function $\ell \in \mathcal {M}(\mathcal {B}_2)$ , we set $a = \ell (e_1)$ , $b = \ell (e_2)$ and $m = \ell (e_1) + \ell (e_2) + 2\ell (e_3)$ . Applying the definition of $\overline {F}_G$ from §4.2 to $\mathcal {B}_2$ , we find the formula

$$ \begin{align*} \overline{F}_{\mathcal{B}_2}(\ell) & = (1 - \exp(-a))(1 - \exp(-b)) - 4\exp(-m). \end{align*} $$

Lemma 6.3. Suppose $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^{1}(\mathcal {B}_{2})$ is a smooth path such that for all $t \in [0,1]$ we have $\dot m_t> 0$ . If $m_0 \geq 4$ , then

$$ \begin{align*} \mathcal{L}_{\mathfrak{h},\mathcal{B}_2}(\ell_t|[0,1]) \geq \frac{1}{\sqrt{2}} (\sqrt{m_1} - \sqrt{m_0}). \end{align*} $$

Proof. Suppose that $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^1(\mathcal {B}_2)$ is a path where $\dot m_t> 0$ as in the statement of the lemma and assume that $m_0 \geq 4$ . We reparametrize the path $\ell _t$ so that $m_t = t$ .

As $\overline {F}_{\mathcal {B}_2}(\ell _t) = 0$ , we have

(6.2) $$ \begin{align} (1-\exp(-a_t))(1 - \exp(-b_t)) = 4\exp(-m_t) = 4\exp(-t). \end{align} $$

This enables us to give an upper bound on the denominator in the expression for the entropy norm. Specifically, using the fact that $x\exp (-x) \leq 1 - \exp (-x)$ for $x \geq 0$ , we have

$$ \begin{align*} \langle \ell_t,\nabla \overline{F}_{\mathcal{B}_2}(\ell_t) \rangle & = a_t\exp(-a_t)(1 - \exp(-b_t))\\ &\quad+ b_t\exp(-b_t)(1 - \exp(-a_t)) + 4m_t\exp(-m_t) \\ & \leq 2(1-\exp(-a_t))(1-\exp(-b_t)) + 4t\exp(-t) \\ & = 4t\exp(-t) + 8\exp(-t) \\ & \leq 8t\exp(-t). \end{align*} $$

The penultimate line follows from (6.2), and in the final inequality we use the fact that $t \geq 4$ .

Next, we get a lower bound on the numerator in the expression for the entropy norm. We claim that $-\langle \dot \ell _t,\mathbf {H}[\overline {F}_{\mathcal {B}_2}(\ell _t)]\dot \ell _t \rangle \geq (\dot m_t)^2\exp (-m_t)$ . We have that

$$ \begin{align*} -\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\mathcal{B}_2}(\ell_t)]\dot \ell_t \rangle & = (\dot a_t)^2\exp(-a_t)(1 - \exp(-b_t)) + (\dot b_t)^2\exp(-b_t)(1 - \exp(-a_t)) \\ & \quad - 2\dot a_t \dot b_t\exp(-a_t - b_t) + 4(\dot m_t)^2\exp(-m_t). \end{align*} $$

Therefore if $\dot a_t \dot b_t < 0$ then each term is positive and so the claim holds. Therefore, we assume that $\dot a_t$ and $\dot b_t$ have the same sign. As $\langle \dot \ell _t , \nabla \overline {F}_{\mathcal {B}_2}(\ell _t) \rangle = 0$ , we have that

$$ \begin{align*} 4\dot m_t\exp(-m_t) = -\dot a_t\exp(-a_t)(1 - \exp(-b_t)) -\dot b_t\exp(-b_t)(1 - \exp(-a_t)). \end{align*} $$

We write this equation as $w = u + v$ . As $\overline {F}_{\mathcal {B}_2}(\ell _t) = 0$ , we find that

$$ \begin{align*} 2uv & = 2\dot a_t \dot b_t\exp(-a_t)\exp(-b_t)(1 - \exp(-a_t))(1 - \exp(-b_t))\\ &= 2\dot a_t \dot b_t\exp(-a_t-b_t)(4\exp(-m_t)). \end{align*} $$

As $2xy \leq \tfrac 34(x+y)^2$ for all x and y, we find that

$$ \begin{align*} 2\dot a_t \dot b_t\exp(-a_t-b_t)(4\exp(-m_t)) & \leq \tfrac{3}{4}(4\dot m_t\exp(- m_t))^2 = 3(\dot m_t)^2(4\exp(-2m_t)). \end{align*} $$

Therefore $2\dot a_t \dot b_t\exp (-a_t-b_t) \leq 3(\dot m_t)^2\exp (-m_t)$ . From this the claim now follows, and furthermore that

$$ \begin{align*}-\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\mathcal{B}_2}(\ell_t)]\dot \ell_t \rangle \geq (\dot m_t)^2\exp(- m_t) = \exp(-t).\end{align*} $$

Hence we find that the entropy norm along this path is bounded below by

$$ \begin{align*} \| {(\ell_t,\dot \ell_t)} \|^2_{\mathfrak{h},\mathcal{B}_2} = \frac{-\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\mathcal{B}_2}(\ell_t)]\dot \ell_t \rangle}{\langle \ell_t,\nabla \overline{F}_{\mathcal{B}_2}(\ell_t) \rangle}\geq \frac{1}{8t}. \end{align*} $$

Therefore the length of this path in the entropy metric is at least

$$ \begin{align*} \int_{m_0}^{m_1} \sqrt{\frac{1}{8t}} \, dt &= \frac{1}{\sqrt{2}} (\sqrt{m_1} - \sqrt{m_0}).\\[-42pt] \end{align*} $$

As for the 2-rose, we obtain the following proposition.

Proposition 6.4. Suppose $\ell $ and $\ell '$ are length functions in $\mathcal {M}^1(\mathcal {B}_2)$ where $m = \ell (e_1) + \ell (e_2) + 2\ell (e_3)$ and $m' = \ell '(e_1) + \ell '(e_2) + 2\ell '(e_3)$ are at least $4$ . Then

$$ \begin{align*} d_{\mathfrak{h},\mathcal{B}_2}(\ell,\ell') \geq \frac{1}{\sqrt{2}} (\sqrt{m'} - \sqrt{m}). \end{align*} $$

6.3 The theta graph

Let $\Theta _2$ denote the graph with vertices v and w, and edges $e_1$ , $e_2$ and $e_3$ where $o(e_i) = v$ and $\tau (e_i) = w$ for $i \in \{1,2,3\}$ . To make the calculations in this section easier to read, given a length function $\ell \in \mathcal {M}(\Theta _2)$ , we set $a = \ell (e_1) + \ell (e_2)$ , $b = \ell (e_2) + \ell (e_3)$ , $c = \ell (e_3) + \ell (e_1)$ and $m = \ell (e_1) + \ell (e_2) + \ell (e_3)$ . Applying the definition of $\overline {F}_G$ from §4.2 to $\Theta _2$ , we find the formula

$$ \begin{align*} \overline{F}_{\Theta_2}(\ell) = 1 - \exp(-a) - \exp(-b) - \exp(-c) - 2\exp(-m). \end{align*} $$

Lemma 6.5. Suppose $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^{1}(\Theta _{2})$ is a smooth path such that for all $t \in [0,1]$ we have $\dot m_t> 0$ . If $m_0 \geq 4$ , then

$$ \begin{align*} \mathcal{L}_{\mathfrak{h},\Theta_2}(\ell_t|[0,1]) \geq \frac{1}{\sqrt{5}} (\sqrt{m_1} - \sqrt{m_0}). \end{align*} $$

Proof. Suppose that $\ell _t \colon \thinspace [0,1] \to \mathcal {M}^1(\Theta _2)$ is a path where $\dot m_t> 0$ as in the statement of the lemma and assume that $m_0 \geq 4$ . We reparametrize the path $\ell _t$ so that $m_t = t$ .

Let $n_t = \min \{a_t,b_t,c_t\}$ . As $\overline {F}_{\Theta _2}(\ell _t) = 0$ , we have

$$ \begin{align*} 1 - 3\exp(-n_t) \leq 1 - \exp(-a_t) - \exp(-b_t) - \exp(-c_t) = 2\exp(-m_t) = 2\exp(-t). \end{align*} $$

In particular, $3\exp (-n_t) \geq 1- 2\exp (-t)> 3\exp (-2)$ as $t \geq 4$ and thus $n_t < 2$ for all t.

Setting $p_t = \max \{a_t,b_t,c_t\}$ and $q_t = a_t + b_t + c_t - p_t - n_t$ so that $\{ p_t,q_t,n_t \} = \{ a_t,b_t,c_t \}$ for all t, we find that $p_t,q_t \geq t-2$ as $p_t + q_t = 2t - n_t \geq 2t - 2$ and ${p_t,q_t \leq t}$ . Therefore, as $t - 2 \geq 2$ for $t \geq 4$ , the cycle that realizes the minimum of $\{a_t,b_t\,c_t\}$ does not depend on t and therefore we may assume that $c_t = \min \{a_t,b_t,c_t\}$ and thus $a_t,b_t \geq t-2$ . Therefore $\exp (-a_t),\exp (-b_t) \leq \exp (-t+2)$ . Hence, again as $\overline {F}_{\Theta _2}(\ell _t) = 0$ , we have

$$ \begin{align*} 1 - \exp(-c_t) &= \exp(-a_t) + \exp(-b_t) + 2\exp(-m_t)\\ &\leq (2\exp(2) + 2)\exp(-t) \leq 20\exp(-t). \end{align*} $$

This enables us to give an upper bound on the denominator in the expression for the entropy norm. Specifically, using the fact that $x\exp (-x) \leq 1 - \exp (-x)$ for $x \geq 0$ , we have

$$ \begin{align*} \langle \ell_t,\nabla \overline{F}_{\Theta_2}(\ell_t) \rangle & = a_t\exp(-a_t) + b_t\exp(-b_t) + c_t\exp(-c_t)+ 2m_t\exp(-m_t) \\ & \leq a_t\exp(-a_t) + b_t\exp(-b_t) + 1 - \exp(-c_t)+ 2m_t\exp(-m_t) \\ & \leq t\exp(-t+2) + t\exp(-t+2) + 20\exp(-t) + 2t\exp(-t) \\ & \leq 20t\exp(-t) + 20\exp(-t) \\ & \leq 40t\exp(-t). \end{align*} $$

In the final inequality we used the fact that $t \geq 4$ .

Next, we get a lower bound on the numerator in the expression for the entropy norm by just using $m_t$ . Specifically,

$$ \begin{align*} -\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\Theta_2}(\ell_t)]\dot \ell_t \rangle & = (\dot a_t)^2\exp(-a_t) + (\dot b_t)^2\exp(-b_t)\\ &\quad+ (\dot c_t)^2\exp(-c_t) + 2(\dot m_t)^2\exp(-m_t) \\ & \geq 2\exp(-t). \end{align*} $$

Hence we find that the entropy norm along this path is bounded below by

$$ \begin{align*} \| {(\ell_t,\dot \ell_t)} \|^2_{\mathfrak{h},\Theta_2} = \frac{-\langle \dot \ell_t,\mathbf{H}[\overline{F}_{\Theta_2}(\ell_t)]\dot \ell_t \rangle}{\langle \ell_t,\nabla \overline{F}_{\Theta_2}(\ell_t) \rangle}\geq \frac{1}{20t}. \end{align*} $$

Therefore the length of this path in the entropy metric is at least

$$ \begin{align*} \int_{m_0}^{m_1} \sqrt{\frac{1}{20t}} \, dt &= \frac{1}{\sqrt{5}} (\sqrt{m_1} - \sqrt{m_0}).\\[-42pt] \end{align*} $$

Again, as for the 2-rose, we obtain the following proposition.

Proposition 6.6. Suppose $\ell $ and $\ell '$ are length functions in $\mathcal {M}^1(\Theta _2)$ where $m = \ell (e_1) + \ell (e_2) + \ell (e_3)$ and $m' = \ell '(e_1) + \ell '(e_2) + \ell '(e_3)$ are at least $4$ . Then

$$ \begin{align*} d_{\mathfrak{h},\Theta_2}(\ell,\ell') \geq \frac{1}{\sqrt{5}} (\sqrt{m'} - \sqrt{m}). \end{align*} $$

6.4 $(\mathcal {X}^1(\mathbb {F}_2),d_{\mathfrak {h}})$ is complete

We can now prove the main result of this section that the entropy metric on $\mathcal {X}^1(\mathbb {F}_2)$ is complete. Given a marked metric graph $x = [(G,\rho ,\ell )]$ in $\mathcal {X}^1(\mathbb {F}_2)$ , we let

$$ \begin{align*}\textsf{m}(x) = \begin{cases} \ell(e_1) + \ell(e_2) & \mbox{if } G = \mathcal{R}_2, \\ \ell(e_1) + \ell(e_2) + 2\ell(e_3) & \mbox{if } G = \mathcal{B}_2, \\ \ell(e_1) + \ell(e_2) + \ell(e_3) & \mbox{if } G = \Theta_2 \\ \end{cases}\end{align*} $$

We remark that $\textsf {m} \colon \thinspace \mathcal {X}^1(\mathbb {F}_2) \to \mathbb {R}$ is an $\operatorname {\mathrm {Out}}(\mathbb {F}_2)$ -invariant continuous function.

Proof. Suppose that $(x_n)_{n \in \mathbb {N}}$ is a sequence in $\mathcal {X}^1(\mathbb {F}_2)$ such that $\textsf {m}(x_n) \to \infty $ . We will show that $(x_n)_{n \in \mathbb {N}}$ is not Cauchy by showing that $\limsup d_{\mathfrak {h}}(x_1,x_n) = \infty $ .

Let $m = \max \{\textsf {m}(x_1),4\}$ . Given $N \geq 0$ , we let n be such that $\sqrt {\textsf {m}(x_n)} \geq \sqrt {5}N + \sqrt {m}$ . Consider a path $\alpha _t \colon \thinspace [0,1] \to \mathcal {X}^1(\mathbb {F}_2)$ such that $\alpha _0 = x_1$ and $\alpha _1 = x_n$ . Let $\delta \in [0,1]$ be the minimal value such that $\textsf {m}(\alpha _t) \geq 4$ for all $t \in [\delta ,1]$ . In particular, $\textsf {m}(\alpha _\delta ) \leq m$ .

Combining Propositions 6.2, 6.4 and 6.6, we find that

$$ \begin{align*} \mathcal{L}_{\mathfrak{h}}(\alpha_t|[\delta,1]) \geq \frac{1}{\sqrt{5}}( \sqrt{\textsf{m}(x_n)} - \sqrt{\textsf{m}(\alpha_\delta)}) \geq \frac{1}{\sqrt{5}}( \sqrt{\textsf{m}(x_n)} - \sqrt{m}) \geq N. \end{align*} $$

This also provides a lower bound on $\mathcal {L}_{\mathfrak {h}}(\alpha _t|[0,1])$ . Since the path was arbitrary, this also is a lower bound on $d_{\mathfrak {h}}(x_1,x_n)$ .

Therefore $d_{\mathfrak {h}}(x_1,x_n) \geq N$ , showing that $\limsup d_{\mathfrak {h}}(x_1,x_n) = \infty $ as claimed.

Given $L \geq 0$ , we set $\mathcal {X}^1_L(\mathbb {F}_2) = \{ x \in \mathcal {X}^1(\mathbb {F}_2) \mid \textsf {m}(x) \leq L \}$ , and additionally, given a marked graph $\rho \colon \thinspace \mathcal {R}_2 \to G$ , we set $\mathcal {X}^1_L(G,\rho ) = \mathcal {X}^1(G,\rho ) \cap \mathcal {X}^1_L(\mathbb {F}_2)$ . We remark that the closure of $\mathcal {X}^1_L(G,\rho )$ is compact. As $\mathcal {X}^1(\mathbb {F}_2)$ is locally finite and there are only finitely many topological types of graphs, the following statement holds. For all $L, D \geq 0$ , there is an N such that if $x \in \mathcal {X}^1_L(\mathbb {F}_2)$ then there is a collection of marked graphs $(G_1,\rho _1),\ldots ,(G_N,\rho _N)$ such that

$$ \begin{align*} \{x' \in \mathcal{X}^1_L(\mathbb{F}_2) \mid d_{\mathfrak{h}}(x,x') \leq D \} \subseteq \bigcup_{k=1}^N \mathcal{X}^1_L(G_k,\rho_k).\end{align*} $$

Proof. Let $(x_n)_{n \in \mathbb {N}}$ be a Cauchy sequence in $(\mathcal {X}^1(\mathbb {F}_2),d_{\mathfrak {h}})$ . By Lemma 6.7, there is an L such that $(x_n) \subset \mathcal {X}^1_L(\mathbb {F}_2)$ . Let $n_0$ be such that $d_{\mathfrak {h}}(x_n,x_m) \leq 1$ if $n,m \geq n_0$ . By the above remark, there are finitely marked graphs $(G_1,\rho _1), \ldots , (G_N,\rho _N)$ such that

$$ \begin{align*} \{ x_n | n \geq n_0 \} \subset \{ x' \in \mathcal{X}^1_L(\mathbb{F}_2) \mid d_{\mathfrak{h}}(x_{n_0},x') \leq 1 \} \subseteq \bigcup_{k=1}^N \mathcal{X}^1_L(G_k,\rho_k). \end{align*} $$

As the closure of this set in $\mathcal {X}^1(\mathbb {F}_2)$ is compact, the sequence $(x_n)_{n \in \mathbb {N}}$ converges.

7.1 $\mathcal {M}^1(\mathcal {R}_r)$ as a zero locus

In this section we compute the function $\overline {F}_{\mathcal {R}_r}$ . This appears as Proposition 7.3. In Proposition 7.5 we prove that $\mathcal {M}^1(\mathcal {R}_r) = \{\ell \in \mathcal {M}(\mathcal {R}_r) \mid \overline {F}_{\mathcal {R}_r}(\ell ) = 0 \}$ ; this strengthens Proposition 4.13 in this setting.

First we set some notation for working with the graph $\mathcal {R}_r$ . We identify the unoriented edges of $\mathcal {R}_{r}$ with the set $[r] = \{1,2,\ldots ,r\}$ . To simplify the expressions, we will use r as the identifying subscript rather than $\mathcal {R}_{r}$ and we will use variables $\ell = (\ell ^1,\ldots , \ell ^r)$ to denote the length of the unoriented edges. The matrix $\overline {A}_{r,\ell } \in \operatorname {\mathrm {Mat}}_{r}(\mathbb {R})$ has rows and columns indexed by $[r]$ , and we have

(7.1) $$ \begin{align} \overline{A}_{r,\ell}(i,j) = \exp(-\ell^i)(2-\delta(i,j)) \end{align} $$

where is the Kronecker delta function.

For the calculations in this section, we need the following combinatorial identities.

Lemma 7.1. For any $r \geq 1$ and any $x \in \mathbb {R}$ , the following equations hold:

(7.2) $$ \begin{align} (1 + x)^{r-1}(x - (2r-1)) &= x^r\sum_{k=0}^{r} \genfrac{(}{)}{0pt}{0}{r}{k} (1-2k) x^{-k}, \end{align} $$

(7.3) $$ \begin{align} (1 + x)^{r-1}(x + (2r+1)) &= x^r\sum_{k=0}^{r} \genfrac{(}{)}{0pt}{0}{r}{k} (1+2k) x^{-k}. \end{align} $$

Proof. Differentiate the equation $(1+x)^{r} = \sum _{k=0}^{r}\genfrac {(}{)}{0pt}{1}{r}{k} x^{r-k}$ and multiply it by x to obtain the equality $rx(1+x)^{r-1} = \sum _{k=0}^{r} (r-k) \genfrac {(}{)}{0pt}{1}{r}{k} x^{r-k}$ . Therefore

$$ \begin{align*} 2rx(1+x)^{r-1} - (2r-1)(1+x)^{r} &= 2r\sum_{k=0}^{r} \genfrac{(}{)}{0pt}{0}{r}{k} x^{r-k} - \sum_{k=0}^{r} 2k \genfrac{(}{)}{0pt}{0}{r}{k} x^{r-k} \\ & \quad - 2r\sum_{k=0}^{r} \genfrac{(}{)}{0pt}{0}{r}{k} x^{r-k} + \sum_{k=0}^{r} \genfrac{(}{)}{0pt}{0}{r}{k} x^{r-k} \\ & = \sum_{k=0}^{r} (1 - 2k) \genfrac{(}{)}{0pt}{0}{r}{k}x^{r-k} = x^{r}\sum_{k=0}^{r}(1-2k)\genfrac{(}{)}{0pt}{0}{r}{k} x^{-k}. \end{align*} $$

The left-hand side in the above equation simplifies to $(1 + x)^{r-1}(x - (2r-1))$ . This shows (7.2). In a similar manner one can derive (7.3).

Corollary 7.2. For any $r \geq 1$ ,

(7.4) $$ \begin{align} \sum_{k=0}^{r} \genfrac{(}{)}{0pt}{0}{r}{k} (1-2k)(2r-1)^{-k} = 0. \end{align} $$

Given a length function $\ell = (\ell ^1,\ldots ,\ell ^n) \in \mathcal {M}(\mathcal {R}_r)$ and a subset $S \subseteq [r]$ , we define $\ell ^S = \sum _{k \in S} \ell ^k$ . In particular, we have $\ell ^\emptyset = 0$ .

Proposition 7.3. For any $r \geq 2$ and any length function $\ell \in \mathcal {M}(\mathcal {R}_r)$ ,

(7.5) $$ \begin{align} \overline{F}_{r}(\ell) = \sum_{S \subseteq [r]} (1 - 2\lvert {S} \rvert)\exp(-\ell^S). \end{align} $$

Proof. Using the expansion of the determinant via permutations of $[r]$ , we can express $\overline {F}_{r}(\ell ) = \det (I - \overline {A}_{r,\ell })$ as

$$ \begin{align*} \overline{F}_{r}(\ell) = \sum_{S \subseteq [r]} c_{S,r}\exp(-\ell^{S}) \end{align*} $$

for some coefficients $c_{S,r} \in \mathbb {R}$ depending on the subset $S \subseteq [r]$ and the rank r. Further, it is apparent that the coefficient $c_{S,r}$ only depends on the cardinality of S. It remains to determine these coefficients. We will do so by induction.

For $r=2$ , we compute

$$ \begin{align*} \overline{F}_{2}(\ell^{1},\ell^{2}) &= \det\left[\begin{matrix} 1 - \exp(-\ell^{1}) & -2\exp(-\ell^{1}) \\ -2\exp(-\ell^{2}) & 1 - \exp(-\ell^{2}) \end{matrix}\right] \\ & = 1 - \exp(-\ell^{1}) - \exp(-\ell^{2}) - 3\exp(-\ell^{1} - \ell^{2}). \end{align*} $$

This shows the proposition for $r = 2$ .

Suppose $r \geq 3$ and that the proposition holds for $r-1$ . That is, we assume that $c_{S,r-1} = 1-2\lvert {S} \rvert $ for any $S \subseteq [r-1]$ . Since $c_{S,r}$ only depends on the cardinality of S, this implies that $c_{S,r} = 1 - 2\lvert {S} \rvert $ for any $S \subseteq [r]$ where $\lvert {S} \rvert < r$ as well. Hence it only remains to compute $c_{[r],r}$ .

To compute $c_{[r],r}$ , we make use of Corollary 7.2. Indeed, by Example 3.3, we have $\mathfrak {h}_r(\log (2r-1) \cdot \mathbf {1}) {{\kern-1pt}={\kern-1pt}} 1$ . Therefore, by Proposition 4.13, we obtain ${\overline {F}_{r}(\log (2r-1) \cdot \mathbf {1}) {{\kern-1pt}={\kern-1pt}} 0}$ . Hence

$$ \begin{align*} 0 & = \overline{F}_{r}(\log(2r-1) \cdot \mathbf{1}) \\ & = \sum_{S \subset [r]} (1 - 2\lvert {S} \rvert) \exp(-\lvert {S} \rvert\log(2r-1)) + c_{[r],r}\exp(-r\log(2r-1)) \\ & = \sum_{k=0}^{r-1} \genfrac{(}{)}{0pt}{0}{r}{k} (1-2k)(2r-1)^{-k} + c_{[r],r}(2r-1)^{-r}. \end{align*} $$

By Corollary 7.2, we find that $c_{[r],r} = 1-2r$ as desired.

Example 7.4. For $r = 2$ and $r = 3$ , using the coordinates $x = \exp (-\ell ^{1})$ , $y = \exp (-\ell ^{2})$ and $z = \exp (-\ell ^{3})$ , we find

$$ \begin{align*} \overline{F}_{2}(\ell^{1},\ell^{2}) &= 1 - x - y - 3xy,\\ \overline{F}_{3}(\ell^{1},\ell^{2},\ell^{3}) &= 1 - x - y - z - 3xy - 3xz - 3yz - 5xyz. \end{align*} $$

Figure 5 shows $\mathcal {M}^{1}(\mathcal {R}_{r})$ as a subset of $\mathcal {M}(\mathcal {R}_r)$ for $r = 2$ and $r = 3$ .

Figure 5

The hypersurfaces $\mathcal {M}^{1}(\mathcal {R}_{r})$ for the roses with 2 and 3 petals.

Using Proposition 7.3, we can provide a strengthening of Proposition 4.13 for the r-rose.

Proof. By Proposition 4.13, we have that $\mathcal {M}^{1}(\mathcal {R}_{r}) \subseteq \{\ell \in \mathcal {M}(\mathcal {R}_r) \mid \overline {F}_{r}(\ell ) = 0 \}$ . Suppose that $\overline {F}_{r}(\ell ) = 0$ for some $\ell \in \mathcal {M}(\mathcal {R}_r)$ . Set $h = \mathfrak {h}_r(\ell )$ . We need to show that $h = 1$ .

Consider the function $p \colon \thinspace \mathbb {R}_{>0} \to \mathbb {R}$ defined by $p(t) = \overline {F}_{r}(t \cdot \ell )$ . We have $p(1) = \overline {F}_{r}(\ell ) = 0$ . As $\mathfrak {h}_r(h \cdot \ell ) = 1$ , we have $p(h) = \overline {F}_{r}(h \cdot \ell ) = 0$ as well by Proposition 4.13.

Using the expression for $\overline {F}_r(\ell )$ derived in Proposition 7.3, we compute that

$$ \begin{align*} p'(t) = \sum_{\substack{S \subseteq [r] \\ S \neq \emptyset}} (2\lvert {S} \rvert - 1)\ell^{S}\exp(-t \cdot \ell^{S}). \end{align*} $$

Therefore $p'(t)> 0$ for all $t \in \mathbb {R}_{>0}$ . As $p(h \cdot \ell ) = 0 = p(\ell )$ , we must have that $h \cdot \ell = \ell $ and hence $h = 1$ .

7.2 Finite-length paths in $\mathcal {M}^1(\mathcal {R}_r)$ for $r \geq 3$

Using the computation of $\overline {F}_r$ , in Proposition 7.8 we will compute the length of the path in $\mathcal {M}^1(\mathcal {R}_r)$ starting at $\log (2r-1) \cdot \mathbf {1}$ that blows up the length of one edge while shrinking the lengths of the others at the same rate. As we will show, when r is at least $3$ , this path has finite length and thus the moduli space $(\mathcal {M}^1(\mathcal {R}_r),d_{\mathfrak {h},\mathcal {R}_r})$ is not complete for $r \geq 3$ .

Before we begin, it is useful to introduce the following functions $X_{i}, Y_{i} \colon \thinspace \mathcal {M}(\mathcal {R}_r) \to \mathbb {R}$ for each $i \in [r]$ :

(7.6) $$ \begin{align} X_{i}(\ell) &= \sum_{S \subseteq [r] - \{i\}} (1 - 2\lvert {S} \rvert) \exp(-\ell^{S}), \end{align} $$

(7.7) $$ \begin{align} Y_{i}(\ell) &= \sum_{S \subseteq [r] - \{i\}} (1 + 2\lvert {S} \rvert) \exp(-\ell^{S}). \end{align} $$

Both $X_{i}$ and $Y_{i}$ are constant with respect to $\ell ^{i}$ . Using these functions, we can isolate the terms in $\overline {F}_r(\ell )$ in which $\ell ^i$ appears and write