2 Strongly reversible elements and common perpendiculars

Let X be

• • either a complete simply connected Riemannian manifold ${\widetilde {M}}$ with dimension at least $2$ and with pinched negative sectional curvature $-a^2\leq K\leq -1$ ,

• • or the geometric realisation of a simplicial tree ${\mathbb X}$ without terminal vertices and with uniformly bounded degrees of vertices.

We denote by $X\cup \partial _\infty X$ the geometric compactification of X, where $\partial _\infty X$ is the boundary at infinity of X. For every closed convex subset A of X, we denote by $\overline {A}$ its closure in $X\cup \partial _\infty X$ .

When X is a manifold, we denote by $\operatorname {Isom}(X)$ its locally compact full isometry group. When X is a tree, we denote by $\operatorname {Isom}(X)$ the locally compact group of automorphisms of ${\mathbb X}$ without edge inversion. Let $\Gamma $ be a nonelementary discrete subgroup of $\operatorname {Isom}(X)$ . An element in $\operatorname {Isom}(X)$ of order $2$ is called an involution. We assume from now on that $\Gamma $ contains involutions. See for instance [Reference Bridson and HaefligerBrH] for background on $\operatorname {CAT}(-1)$ spaces and their discrete groups of isometries, and [Reference SerreSer] for background on group actions on trees, as well as [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Chap. 2].

For every $\gamma \in \operatorname {Isom}(X)$ , we denote by

$$\begin{align*}\lambda(\gamma)=\inf_{x\in X} d(x,\gamma x) \end{align*}$$

the translation length of $\gamma $ . The element $\gamma $ is loxodromic if $\lambda (\gamma )>0$ . We then denote its translation axis by

$$\begin{align*}\operatorname{Ax}_\gamma=\{x\in X:d(x,\gamma x)=\lambda(\gamma)\}\,, \end{align*}$$

and its repelling, attracting fixed points at infinity by $\gamma _-,\gamma _+ \in \partial _\infty X$ respectively. The element $\gamma $ is elliptic if it has a fixed point in X and parabolic if it is neither loxodromic nor elliptic.

The discrete group $\Gamma $ acts by conjugation on the nonempty set $I_\Gamma $ of involutions of $\Gamma $ . In what follows, I and J will denote two fixed, $\Gamma $ -invariant and nonempty subsets of $I_\Gamma $ , and will be endowed with the left action by conjugation of $\Gamma $ . For instance, given an involution $\alpha $ of $\Gamma $ , the set of the conjugates of $\alpha $ by the elements of $\Gamma $ will be denoted by $I_\alpha $ .

Let $\alpha \in I_\Gamma $ . We denote by

$$\begin{align*}F_\alpha=\{x\in X:\alpha x =x\} \end{align*}$$

the fixed point set of $\alpha $ , which is a proper, nonempty,Footnote ¹ closed and convexFootnote ² subset of X. It is a totally geodesic submanifold when X is a manifold by for instance [Reference Gallot, Hulin and LafontaineGHL, §2.80 bis], and the geometric realisation of a simplicial subtree of ${\mathbb X}$ when X is a tree.

An element $\gamma \in \Gamma $ is reversible in $\Gamma $ if it is conjugated to its inverse. It is strongly reversible in $\Gamma $ if it is conjugated to its inverse by an involution in $\Gamma $ . We refer to [Reference O’Farrell and ShortO’FS, Sect. 2] for the basic ideas of reversibility, and the rest of the cited book for an extensive survey. In particular, an element of $\Gamma $ is strongly reversible if and only if it is the product of two involutions of $\Gamma $ , see [Reference O’Farrell and ShortO’FS, Prop. 2.12]. An element $\gamma \in \Gamma $ is $\{I,J\}$ -reversible if there exists an element $\alpha \in I$ such that we have $\gamma \alpha =\alpha \gamma ^{-1} \in J$ . Such an element $\alpha \in I_{\Gamma }$ is called a $\gamma $ -reversing involution for $(I,J)$ .

In this paper, we are interested in strongly reversible and loxodromic elements of $\Gamma $ . We denote by ${\widetilde {\mathfrak R}}_{I,J}$ the set of $\{I,J\}$ -reversible loxodromic elements of $\Gamma $ . Note that if $I'$ and $J'$ are $\Gamma $ -invariant subsets of $I_\Gamma $ such that $I\subset I'$ and $J\subset J'$ , then ${\widetilde {\mathfrak R}}_{I,J}\subset {\widetilde {\mathfrak R}}_{I',J'}$ .

Let $D^-$ and $D^+$ be two nonempty, proper, closed and convex subsets of X. A geodesic arc ${\widetilde {\rho }}: [0,T]\rightarrow X$ , where $T>0$ , ${\widetilde {\rho }}(0)\in D^-$ and ${\widetilde {\rho }}(T)\in D^+$ is a common perpendicular of length $\lambda ({\widetilde {\rho }})=T$ from $D^-$ to $D^+$ if its image is the unique shortest geodesic segment from a point of $D^-$ to a point of $D^+$ . It exists if and only if the closures $\overline {D^-}$ and $\overline {D^+}$ of $D^-$ and $D^+$ in $X\cup \partial _\infty X$ are disjoint. We refer to [Reference Parkkonen and PaulinPP3], the survey [Reference Parkkonen and PaulinPP2], or [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, §2.5, §12.1 and §12.4] for further details on common perpendiculars.

The following lemma connects strongly reversible loxodromic elements of $\Gamma $ with common perpendiculars between the fixed point sets of involutions. This connection is the key to the proofs of the main results in Sections 5 and 6.

Lemma 2.3. Let $(\alpha ,\beta )\in I_\Gamma \times I_\Gamma $ , and let I and J be two $\Gamma $ -invariant and nonempty subsets of $I_\Gamma $ . The product $\gamma =\beta \alpha $ is strongly reversible, and it is $\{I,J\}$ -reversible if furthermore $\alpha \in I$ and $\beta \in J$ . It is

1. (i) elliptic or the identity if and only if $F_\alpha \cap F_\beta \neq \emptyset $ ,

2. (ii) loxodromic if and only if $\overline {F_\alpha } \,\cap \,\overline { F_\beta } =\emptyset $ ,

3. (iii) parabolic if and only if $F_\alpha \cap F_\beta =\emptyset $ and $\overline {F_\alpha }\,\cap \,\overline {F_\beta }\neq \emptyset $ .

In cases (ii) and (iii), the group $\langle \alpha ,\beta \rangle $ generated by $\alpha $ and $\beta $ is an infinite dihedral group and we have

$$\begin{align*}\langle\alpha,\beta\rangle=\langle\alpha\rangle*\langle\beta\rangle\;. \end{align*}$$

If $\gamma $ is loxodromic, the translation axis $\operatorname {Ax}_\gamma $ of $\gamma $ is the union of the images by the elements of the group $\langle \alpha , \beta \rangle $ of the common perpendicular segment ${\widetilde {\rho }}_{\alpha , \beta }$ from $F_{\alpha }$ to $F_{\beta }$ , and the translation length of $\gamma $ satisfies

(1) $$ \begin{align} \lambda(\gamma)=2\;d(F_\alpha,F_\beta)\,. \end{align} $$

Proof. Let $\alpha \in I$ and $\beta \in J$ (with possibly $I=J= I_\Gamma $ ). Let $\gamma =\beta \alpha $ . Then $\alpha $ is a $\gamma $ -reversing involution for $(I,J)$ , and hence $\gamma $ is $\{I,J\}$ -reversible, since $\gamma \alpha =\beta \in J$ and

(2) $$ \begin{align} \alpha \gamma\alpha^{-1}= \alpha (\beta \alpha ) \alpha^{-1}=\alpha \beta=(\beta \alpha )^{-1}=\gamma^{-1}\,. \end{align} $$

If $F_\alpha \cap F_\beta \neq \emptyset $ , then $\gamma =\beta \alpha $ pointwise fixes this intersection and $\gamma $ is elliptic.

Assume that $F_\alpha \cap F_\beta =\emptyset $ . If the intersection $\overline {F_\alpha } \,\cap \,\overline {F_\beta }$ is nonempty, then by convexity, this intersection is reduced to a point at infinity $\xi \in \partial _\infty X$ , which is fixed by $\alpha $ and $\beta $ , hence by $\gamma $ . If X is a tree, then the two convex subsets $F_\alpha $ and $F_\beta $ would meet, which has been excluded. Hence X is a manifold. Any horosphere centred at $\xi $ is invariant by $\alpha $ and $\beta $ , hence by $\gamma =\beta \alpha $ and by $\gamma ^{-1}= \alpha \beta $ , so that $\gamma $ is parabolic or elliptic. If a point $x\in X$ is fixed by $\beta \alpha $ , then $\beta $ sends the segment $[x,\alpha x]$ to the segment $[\beta x,x]$ , hence it sends the midpoint $m_\alpha $ of $[x,\alpha x]$ to the midpoint $m_\beta $ of $[\beta x,x]$ . But $m_\alpha \in F_\alpha $ and $m_\beta \in F_\beta $ . Hence $m_\alpha =\beta ^{-1}m_\beta =m_\beta \in F_\alpha \cap F_\beta $ , contradicting the disjointness of $F_\alpha $ and $F_\beta $ . Thus $\gamma $ is not elliptic, therefore it is parabolic. In particular, $\gamma $ has infinite order, and $\langle \alpha , \beta \rangle $ is indeed isomorphic to $\langle \alpha \rangle *\langle \beta \rangle $ .

If $\overline {F_\alpha } \,\cap \,\overline { F_\beta } =\emptyset $ , let ${\widetilde {\rho }}_{\alpha ,\beta }=[x,y]$ be the common perpendicular from $F_\alpha $ to $F_\beta $ (with $x\in F_\alpha $ and $y\in F_\beta $ , see Figure 3). We claim that $\alpha [y,x]\cup [x,y]$ is a geodesic segment. When X is a manifold, this follows from the properties of the action of $T_x\alpha $ on $\nu _xF_\alpha $ described in Remark 2.2 (5). When X is a tree, the interior of $[x,y]$ would otherwise contain a fixed point of $\alpha $ . Similarly, $[\alpha y,y]\cup \beta [y,\alpha y]$ is a geodesic segment, and $\gamma =\beta \alpha $ maps its first half $[\alpha y,y]$ to its second half $[\beta y,\beta \alpha y]$ in an orientation preserving way. This implies that $\gamma $ is loxodromic, that its translation axis contains $[\alpha y,y]$ , hence contains ${\widetilde {\rho }}_{\alpha ,\beta }$ , and that

$$\begin{align*}\lambda(\gamma)=d(\alpha y,y)=2 \,d(x,y)=2\,d(F_\alpha,F_\beta)\,. \end{align*}$$

Taking the image of the common perpendicular segment ${\widetilde {\rho }}_{\alpha ,\beta }$ by $\alpha $ , and then the images of $\alpha {\widetilde {\rho }}_{\alpha ,\beta }\cup {\widetilde {\rho }}_{\alpha ,\beta }$ by the powers of $\gamma $ , we cover the whole translation axis $\operatorname {Ax}_\gamma $ . The fact that $\langle \alpha ,\beta \rangle $ is isomorphic to $\langle \alpha \rangle *\langle \beta \rangle $ follows as in the parabolic case.

Figure 3.

Loxodromic products of two order $2$ elements of $\Gamma $ .

A subgroup H of $\Gamma $ isomorphic to the free product ${\mathbb Z}/2{\mathbb Z}*{\mathbb Z}/2{\mathbb Z}$ of two copies of the cyclic group of order $2$ is an infinite dihedral subgroup of $\Gamma $ . Note that H is isomorphic to the semidirect product ${\mathbb Z}\rtimes {\mathbb Z}/2{\mathbb Z}$ and it has a canonical morphism onto ${\mathbb Z}/2{\mathbb Z}$ (mapping any element of order $2$ to $1\in {\mathbb Z}/2{\mathbb Z}$ ), with kernel a canonical infinite cyclic (normal) subgroup $Z_H$ . If $\alpha \in I$ , $\beta \in J$ and $H=\langle \alpha \rangle *\langle \beta \rangle $ , then $\gamma =\beta \alpha $ generates $Z_H$ , and $\gamma $ is $\{I,J\}$ -reversible in $\Gamma $ with $\alpha $ a $\gamma $ -reversing involution for $(I,J)$ by Lemma 2.3.

We say that an infinite dihedral subgroup H of $\Gamma $ is of loxodromic type if $Z_H$ is generated by a loxodromic element and that it is of parabolic type if $Z_H$ is generated by a parabolic element. Note that when X is a manifold with dimension $2$ and when $\Gamma $ preserves an orientation, then all infinite dihedral subgroups of $\Gamma $ are of loxodromic type. In case (ii) of Lemma 2.3, the group $\langle \alpha ,\beta \rangle $ generated by $\alpha $ and $\beta $ is an infinite dihedral group of loxodromic type. In case (iii), it is an infinite dihedral group of parabolic type.

An $\{I,J\}$ -dihedral subgroup of $\Gamma $ is an infinite dihedral subgroup $\langle \alpha \rangle *\langle \beta \rangle $ of loxodromic type with $\alpha \in I$ and $\beta \in J$ . We denote by ${\widetilde {\mathfrak D}}_{I,J}$ the set of $\{I,J\}$ -dihedral subgroups of $\Gamma $ . It is invariant under conjugation by elements of $\Gamma $ .

We refer to [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, §7.2] for the following definitions concerning equivariant families of (convex) subsets of X. Let $I'$ be an index set endowed with a left action of $\Gamma $ . A family ${\mathcal {D}}=(D_i)_{i\in I'}$ of subsets of X indexed by $I'$ is $\Gamma $ -equivariant if $\gamma D_i=D_{\gamma i}$ for all $\gamma \in \Gamma $ and $i\in I'$ . We denote by the equivalence relation on $I'$ defined by if and only if $D_i=D_j$ and there exists $\gamma \in \Gamma $ such that $j=\gamma i$ . We say that ${\mathcal {D}}$ is locally finite if for every compact subset K in X, the quotient set is finite.

The family ${\mathcal {F}}_I=(F_\alpha )_{\alpha \in I}$ is $\Gamma $ -equivariant since $\gamma F_\alpha =F_{\gamma \alpha \gamma ^{-1}}$ for all $\gamma \in \Gamma $ and $\alpha \in I$ , and locally finite by the discreteness of $\Gamma $ . We simplify the notation as , with

(3)

When X is a manifold, by Remark 2.1 (2), we have if and only if $\alpha = \beta $ .

The equivalence relation on the set I is $\Gamma $ -equivariant: for all $\alpha ,\beta \in I$ and $\gamma \in \Gamma $ , we have if and only if . We henceforth endow the quotient set with the quotient left action of the group $\Gamma $ .

For every $\alpha \in I_\Gamma $ , let $\Gamma _{F_\alpha }=\operatorname {Stab}_\Gamma (F_\alpha )$ be the (global) stabiliser of $F_\alpha $ in $\Gamma $ . Let $Z_\Gamma (\alpha )$ be the centraliser of $\alpha $ in $\Gamma $ , which is contained in $\Gamma _{F_\alpha }$ .

Proof. Let $\gamma \in {\widetilde {\mathfrak R}}_{I,J}$ . By definition, there exists $\alpha \in I$ such that $\beta =\gamma \alpha \in J$ , hence we have $\gamma =\beta \alpha ^{-1}=\beta \alpha $ . For every $\delta \in Z_\Gamma (\gamma )$ , we have

$$\begin{align*}\gamma=\delta\gamma\delta^{-1}= \delta\beta \alpha \delta^{-1}=(\delta\beta\delta^{-1}) (\delta\alpha \delta^{-1})\,, \end{align*}$$

which proves the first claim since I and J are invariant under conjugation.

To prove the second claim, let us fix $x\in \operatorname {Ax}_\gamma $ . Since $Z_\Gamma (\gamma )$ contains $\gamma ^{\mathbb Z}$ which acts on $\operatorname {Ax}_\gamma $ with fundamental domain equal to the relatively compact geodesic segment $[x,\gamma x[\,$ , up to conjugating $\alpha $ by an element $Z_\Gamma (\gamma )$ , we may assume that the unique fixed point $f_{\alpha , \gamma }$ of $\alpha $ on $\operatorname {Ax}_\gamma $ belongs to $[x,\gamma x]$ . The claim follows since the family ${\mathcal {F}}_I$ is locally finite.

Note that when X is a tree, we can have $|\Gamma _{F_\alpha }/ Z_\Gamma (\alpha )|>1$ , see Lemma 7.9. However, when X is a manifold, we have $|\Gamma _{F_\alpha }/ Z_\Gamma (\alpha )|=1$ and the equivalence classes for are singletons, see Remark 2.1 (2).

3 Multisets of strongly reversible loxodromic elements

A set with multiplicity or a (positive real valued) multiset is a pair $A=(\underline A,\omega )$ , where $\underline A$ is a set, called the underlying set of A, and $\omega : \underline A\to \;]0,+\infty [$ is a positive function, called the multiplicity of A. By an element of A, we mean an element of $\underline A$ . If $E=(\underline E,\omega )$ and $E'= (\underline E',\omega ')$ are multisets, a multiset bijection $f:E\to E'$ is a surjective map $f:\underline E\rightarrow \underline E'$ such that for every $x'\in \underline{E'}$ , we have

$$ \begin{align*}\omega'(x')= \sum_{x\in f^{-1} (x')} \omega(x)\,.\end{align*} $$

If $E=(\underline E,\omega )$ is a multiset and $\underline A\subset \underline E$ , the multiset $A=(\underline A,\omega |_A)$ is a restriction of E.

As a preparation for the proofs in Sections 5 and 6, we define various multisets associated with strongly reversible loxodromic elements of $\Gamma $ . At the end of this Section, we prove that there are multiset bijections between these multisets.

Let

$$\begin{align*}{\widetilde{IJ}}=\big\{(\alpha,\beta)\in I\times J : \overline{F_{\alpha}}\,\cap\,\overline{F_{\beta}}\,=\emptyset\big\}\,. \end{align*}$$

Note that $(\alpha ,\beta )\in {\widetilde {IJ}}$ if and only if $(\beta ,\alpha ) \in {\widetilde {JI}}$ . Let be the diagonal equivalence relation on $I\times J$ , defined by if and only if and . The condition $\overline {F_{\alpha }}\,\cap \, \overline {F_{\beta }}\, =\emptyset $ for $(\alpha , \beta )\in I\times J$ is constant on the equivalence classes modulo and on the orbits of the diagonal left action of the group $\Gamma $ on $I \times J$ . Hence the subset ${\widetilde {IJ}}$ of $I\times J$ is saturated by and $\Gamma $ -invariant.

We define the multiset $IJ$ with underlying set $\underline {IJ}= \Gamma \backslash {\widetilde {IJ}}$ and multiplicity given by

(4) $$ \begin{align} \forall\;\Gamma(\alpha,\beta)\in \underline{IJ},\quad \operatorname{mult}(\Gamma(\alpha,\beta))= \frac{1}{{\operatorname{Card}}(\Gamma_{F_\alpha}\cap\Gamma_{F_\beta})\; |\Gamma_{F_\alpha}/Z_\Gamma(\alpha)|\; |\Gamma_{F_\beta}/Z_\Gamma(\beta)|}\,. \end{align} $$

The right hand side of the above formula for $(\alpha , \beta )\in {\widetilde {IJ}}$ is constant on the equivalence classes modulo and on the orbits of $\Gamma $ on ${\widetilde {IJ}}$ , hence $\operatorname {mult}(\Gamma (\alpha ,\beta ))$ is well defined.

For every $T>0$ , we define the multiset restriction $IJ(T)$ of $IJ$ by setting

(5) $$ \begin{align} \underline{IJ(T)}= \big\{\Gamma(\alpha,\beta)\in \underline{IJ}:d(F_\alpha,F_\beta)\leq T\}\,. \end{align} $$

We denote by ${\mathfrak R}_{I,J}$ the multiset, whose underlying set is the set $\underline {{\mathfrak R}_{I,J}}=\Gamma \backslash {\widetilde {\mathfrak R}}_{I, J}$ of conjugacy classes $[\gamma ]$ of $\{I,J\}$ -reversible loxodromic elements $\gamma $ of $\Gamma $ , and where the multiplicity of $[\gamma ]\in {\mathfrak R}_{I,J}$ is defined by

(6) $$ \begin{align} &\operatorname{mult}([\gamma])= \sum_{\Gamma(\alpha,\,\beta)\,\in\,\underline{IJ}\;:\;\beta\alpha\,\in\,[\gamma]}\; \operatorname{mult}(\Gamma(\alpha,\beta))\,. \end{align} $$

We have ${\mathfrak R}_{I,J}={\mathfrak R}_{J,I}$ . For every $T>0$ , we denote by ${\mathfrak R}_{I,J}(T)$ the multiset restriction of ${\mathfrak R}_{I,J}$ to its elements with translation length at most T.

Here is a particular case when the computation of the multiplicity of the conjugacy class of a strongly reversible loxodromic element of $\Gamma$ is easy. An element $\gamma \in \Gamma $ is primitive if it is not a proper power in $\Gamma $ . An element $\gamma \in {\widetilde {\mathfrak R}}_{I,J}$ is $\{I,J\}$ -primitive if it is not a proper power of an element of ${\widetilde {\mathfrak R}}_{I,J}$ . This property being invariant under conjugation by $\Gamma $ , we also say that the conjugacy class $[\gamma ]\in {\mathfrak R}_{I,J}$ of $\gamma $ is $\{I,J\}$ -primitive. We denote the set of conjugacy classes of $\{I,J\}$ -primitive elements in $\Gamma $ by ${\mathfrak R}_{I,\,J,\,\mathrm {prim}}$ .

Lemma 3.2. Assume that ${\widetilde {M}}$ has dimension $2$ , that $\Gamma $ preserves an orientation of ${\widetilde {M}}$ , and that $I=J=I_\Gamma $ . If $\gamma $ is a primitive, strongly reversible and loxodromic element of $\Gamma $ , then

$$\begin{align*}\operatorname{mult}([\gamma])=2\,. \end{align*}$$

Proof. Let $(\alpha ,\,\beta )$ and $(\alpha ',\,\beta ')$ be two elements of ${\widetilde {IJ}}$ such that $[\beta \alpha ]=[\beta '\alpha ']=[\gamma ]$ . Up to replacing them by other elements in their $\Gamma $ -orbits in ${\widetilde {IJ}}$ , we may assume that $\beta \alpha =\beta '\alpha '=\gamma $ . Let $x_\alpha , x_\beta , x_{\alpha '}, x_{\beta '}$ be the (isolated) fixed point of $\alpha ,\beta ,\alpha ', \beta '$ respectively. They belong to the translation axis $\operatorname {Ax}_\gamma $ of $\gamma $ , see Figure 4. Since $[x_\alpha ,\gamma x_\alpha [$ is a fundamental domain for the action of $\gamma ^{\mathbb Z}$ on $\operatorname {Ax}_\gamma $ , up to diagonally conjugating $(\alpha ',\beta ')$ by an element of $Z_\Gamma (\gamma )$ , we may assume that $x_{\alpha '}\in [x_\alpha , \gamma x_\alpha [\,$ .

Figure 4.

The translation axis of a loxodromic strongly reversible element of $\Gamma $ .

If $x_{\alpha '}=x_\alpha $ , then since ${\widetilde {M}}$ is a manifold, we have $\alpha '=\alpha $ by Remark 2.1 (2). Hence, $\beta '=\beta $ and $\Gamma (\alpha ,\,\beta )= \Gamma (\alpha ',\,\beta ')$ . Otherwise, we have the following three cases to consider.

If $x_{\alpha '}\in \;]x_\alpha , x_\beta [\,$ , then the element $\alpha ' \alpha \in \Gamma $ is loxodromic, with same translation axis as $\gamma $ , but with translation length $\lambda (\alpha ' \alpha) = 2\,d(x_\alpha , x_{\alpha '})<2\,d(x_\alpha ,x_\beta )=\lambda (\gamma )$ , contradicting the fact that $\gamma $ is primitive.

If $x_{\alpha '}\in \;]x_\beta ,\gamma x_\alpha [\,$ , then similarly considering the element $\alpha '\beta $ contradicts the fact that $\gamma $ is primitive.

Hence $x_{\alpha '}=x_\beta $ . Therefore $\alpha '=\beta $ , and $x_{\beta '}=\gamma x_\alpha =\beta x_\alpha =x_{\beta \alpha \beta ^{-1}}$ , so that we have $\beta '=\beta \alpha \beta ^{-1}$ , and there is only one possibility that $\Gamma (\alpha ',\, \beta ')= \Gamma (\beta ,\alpha )$ . Note that we have $\Gamma (\alpha ,\,\beta )\neq \Gamma (\beta ,\,\alpha )$ : Otherwise there exists $\delta \in \Gamma $ so that $\beta =\delta \alpha \delta ^{-1}$ and $\alpha = \delta \beta \delta ^{-1}$ . But then $\delta x_\alpha =x_\beta $ and $\delta x_\beta =x_\alpha $ , so that $\delta $ fixes the midpoint of the geodesic segment $[x_\alpha , x_\beta ]$ . Considering the element $\delta \alpha $ again contradicts the fact that $\gamma $ is primitive. This proves the result.

A common perpendicular $\rho $ of type $(I,J)$ is the $\Gamma $ -orbit of the common perpendicular ${\widetilde {\rho }}_{\alpha ,\beta }$ from $F_{\alpha }$ to $F_{\beta }$ , where $(\alpha , \beta )$ ranges over ${\widetilde {IJ}}$ . Note that

(7)

(8) $$ \begin{align} \text{and}\qquad \forall\,(\alpha,\beta)\in {\widetilde{IJ}},\quad\forall\;\gamma\in\Gamma,\qquad \gamma \;{\widetilde{\rho}}_{\alpha,\beta} ={\widetilde{\rho}}_{\gamma\alpha\gamma^{-1},\gamma\beta\gamma^{-1}} \,. \end{align} $$

We then denote , where is defined in Section 4. We remark that such a pair $(\alpha , \beta )$ is not necessarily unique. The length $\lambda (\rho )$ of $\rho $ is the length of any such ${\widetilde {\rho }}_{\alpha , \beta }$ . The multiplicity of $\rho $ is

(9) $$ \begin{align} \operatorname{mult}(\rho) =\sum_{\Gamma(\alpha,\,\beta)\,\in\, \underline{IJ}, \;\;\rho= \rho_{\alpha,\beta}} \frac{1}{{\operatorname{Card}}(\Gamma_{F_\alpha}\cap\Gamma_{F_\beta})\; |\Gamma_{F_\alpha}/Z_\Gamma(\alpha)|\; |\Gamma_{F_\beta}/Z_\Gamma(\beta)|}\,. \end{align} $$

This sum does not depend on the choice of a representative $(\alpha , \beta )$ in each orbit of $\Gamma $ in ${\widetilde {IJ}}$ , and is finite by the local finiteness of the families ${\mathcal {F}}_I$ and ${\mathcal {F}}_J$ . Recall that if $(\alpha ,\beta )\in {\widetilde {IJ}}$ , then $(\beta ,\alpha )\in {\widetilde {JI}}$ , and note that

(10) $$ \begin{align} \operatorname{mult}(\rho_{\alpha,\beta})=\operatorname{mult}(\rho_{\beta,\alpha})\,. \end{align} $$

We denote by $\operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J)$ the multiset of the common perpendiculars of type $(I,J)$ . For every $T>0$ , we denote by $\operatorname {Perp} ({\mathcal {F}}_I,{\mathcal {F}}_J,T)$ the multiset restriction of $\operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J)$ to its elements with length at most T and by $\underline {\operatorname {Perp}} ({\mathcal {F}}_I,{\mathcal {F}}_J,T)$ its underlying set.

Proposition 3.3. There exist two multiset bijections $\Theta _1: IJ\to \operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J)$ and $\Theta _2: IJ\to {\mathfrak R}_{I,J}$ such that for every $(\alpha ,\beta )\in {\widetilde {IJ}}$ , we have

$$\begin{align*}\Theta_1(\Gamma (\alpha,\beta)) = \rho_{\alpha,\beta} \quad\text{and}\quad\Theta_2(\Gamma(\alpha,\beta))=[\beta\alpha]\,. \end{align*}$$

Furthermore, for every $T>0$ , the maps $\Theta _1$ and $\Theta _2$ send $IJ(T)$ to $\operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J,T)$ and ${\mathfrak R}_{I,J}(2T)$ respectively.

Proof. The map $\Theta _1: \underline {IJ}=\Gamma \backslash {\widetilde {IJ}}\to \underline {\operatorname {Perp}}({\mathcal {F}}_I,{\mathcal {F}}_J)$ defined by $\Gamma (\alpha ,\beta )\mapsto \rho _{\alpha ,\beta }=\Gamma {\widetilde {\rho }}_{\alpha ,\beta }$ is well-defined by Equation (8) and is surjective by the definition of a common perpendicular of type $(I,J)$ . By Equations (9) and (4), for every element $\rho \in \operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J)$ , we have $\operatorname {mult}(\rho ) = \sum _{\Gamma (\alpha ,\,\beta )\,\in \,\Theta _1^{-1}(\rho )}\operatorname {mult}(\alpha ,\beta )$ . Hence $\Theta _1$ is a multiset bijection from $IJ$ to $\operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J)$ .

The map ${\widetilde {\Theta }}_2:{\widetilde {IJ}}\to {\widetilde {\mathfrak R}}_{I,J}$ defined by

$$\begin{align*}{\widetilde{\Theta}}_2:(\alpha ,\beta )\mapsto\beta \alpha \end{align*}$$

indeed has values in the set ${\widetilde {\mathfrak R}}_{I,J}$ of $\{I,J\}$ -reversible loxodromic elements of $\Gamma $ by Lemma 2.3. The map ${\widetilde {\Theta }}_2$ is equivariant for the left actions of $\Gamma $ by diagonal conjugation on ${\widetilde {IJ}}$ and by conjugation on ${\widetilde {\mathfrak R}}_{I,J}$ . The map ${\widetilde {\Theta }}_2$ is surjective by Lemma 2.5. Hence ${\widetilde {\Theta }}_2$ induces a surjective quotient map $\Theta _2: \underline {IJ} \rightarrow \underline {{\mathfrak R}_{I,J}}$ defined by $\Gamma (\alpha ,\beta )\mapsto [\beta \alpha ]$ . By Equation (6), we have $\operatorname {mult}([\gamma ])= \sum _{\Gamma (\alpha ,\,\beta )\,\in \,(\Theta _2)^{-1}([\gamma ])} \operatorname {mult}\Gamma (\alpha , \beta )$ . Hence $\Theta _2$ is a multiset bijection from $IJ$ to ${\mathfrak R}_{I,J}$ .

For all $(\alpha ,\beta )\in {\widetilde {IJ}}$ and $T>0$ , the common perpendicular ${\widetilde {\rho }}_{\alpha ,\beta }$ from $F_{\alpha }$ to $F_{\beta }$ has length $\lambda (\rho _{\alpha ,\beta })=d(F_\alpha , F_\beta )$ at most T if and only if the $\{I,J\}$ -reversible loxodromic element $\beta \alpha $ has translation length at most $2T$ , by Equation (1). This proves the last claim of Proposition 3.3.

For every $\{I,J\}$ -dihedral subgroup H of $\Gamma $ , if $\gamma $ generates the infinite cyclic group $Z_H$ , we define the multiplicity of H as

(11) $$ \begin{align} \operatorname{mult}(H)=\operatorname{mult}([\gamma])\,. \end{align} $$

This multiplicity does not depend on the choice of one of the two generators $\gamma $ of $Z_H$ since $[\gamma ]=[\gamma ^{-1}]$ and it is constant on the conjugacy class of H. We denote by ${\mathfrak D}_{I,J}$ the multiset of conjugacy classes in $\Gamma $ of $\{I,J\}$ -dihedral subgroups of $\Gamma $ , whose underlying set is $\Gamma \backslash {\widetilde {\mathfrak D}}_{I,J}$ (see the paragraph before Example 2.4).

The map $\Theta _3$ restricts to a multiset bijection between the multiset of conjugacy classes of maximalFootnote ⁶ $\{I,J\}$ -dihedral subgroups of loxodromic type and the multiset ${\mathfrak R}_{I,\,J,\,\mathrm {prim}}$ of conjugacy classes of $\{I,J\}$ -primitive, $\{I,J\}$ -reversible and loxodromic elements of $\Gamma $ .

4 Dynamics of the geodesic flow

This whole section consists of background information from [Reference Paulin, Pollicott and SchapiraPPS, Chap. 3, 6, 7] and [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Chap. 3, 4], to which we refer for proofs, details and complements, in particular for the definitions of the measures generalising the ones given in the introduction.

Geodesics and the geodesic flow. Let $(X,\Gamma )$ be as in the beginning of Section 2. Let ${\mathcal {G}} X$ be the metric space of all geodesic lines $\ell : {\mathbb R}\rightarrow X$ in X, such that, when X is a tree, $\ell (0)$ is a vertex of ${\mathbb X}$ . The distance between two elements $\ell $ and $\ell '$ in ${\mathcal {G}} X$ is

(12) $$ \begin{align} d(\ell,\ell')= \int_{-\infty}^{\infty} d(\ell(t),\ell'(t))\;e^{-2\,|t|}\;dt\,. \end{align} $$

When $X={\widetilde {M}}$ is a manifold, we identify the unit tangent bundle $T^1{\widetilde {M}}$ with ${\mathcal {G}} X$ by the map that associates to a unit tangent vector $v\in T^1{\widetilde {M}}$ the unique geodesic line $\ell \in {\mathcal {G}} X$ whose tangent vector $\dot \ell (0)$ at time $0$ is v. When $T^1{\widetilde {M}}$ is endowed with Sasaki’s metric, this mapping is an $\operatorname {Isom}({\widetilde {M}})$ -equivariant bi-Hölder-continuous homeomorphism, see [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, §3.1]. We denote by $\iota : {\mathcal {G}} X\rightarrow {\mathcal {G}} X$ the involutive time-reversal map $\ell \mapsto \{t\mapsto \ell (-t)\}$ , and again by $\iota $ the induced map from the phase space $\Gamma \backslash {\mathcal {G}} X$ to itself. The geodesic flow on ${\mathcal {G}} X$ is the $\operatorname {Isom}(X)$ -equivariant one-parameter group of homeomorphisms

$$\begin{align*}{\mathtt{g}^{t}} :\ell\mapsto \{s \mapsto \ell(s+t)\} \end{align*}$$

for all $\ell \in {\mathcal {G}} X$ , with continuous time parameter $t\in {\mathbb R}$ if X is a manifold, and discrete time parameter $t\in {\mathbb Z}$ if X is a tree. We also denote by $({\mathtt {g}^{t}})_t$ the quotient flow on $\Gamma \backslash {\mathcal {G}} X$ , and call it the geodesic flow on $\Gamma \backslash {\mathcal {G}} X$ .

A $1$ -Lipschitz map $w:{\mathbb R}\to X$ which is isometric on a closed interval and has a constant value on each complementary component is a generalised geodesic. If X is a tree, we furthermore require that $w(0)$ and the constant values on the above-mentioned complementary components are vertices of ${\mathbb X}$ . We denote the Bartels-Lück metric space of generalised geodesics by , with the metric given by Equation (12), see [Reference Bartels and LückBaL] and [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, §2.2]. The space isometrically contains ${\mathcal {G}} X$ . The natural action of $\operatorname {Isom}(X)$ on is isometric. The geodesic flow on ${\mathcal {G}} X$ extends by the same formula to a flow on , that we also call the geodesic flow on , and we use the same notation $({\mathtt {g}^{t}})_t$ for it.

We consider a common perpendicular (with endpoints in $V\mathbb{X}$ when X is a tree) between two nonempty convex subsets of X of length T as a generalised geodesic in , being constant on $]-\infty ,0]$ and $[T,+\infty [\,$ .

Potentials. We now introduce the supplementary data (with origin in physics) of potentials on X. Assume first that $X={\widetilde {M}}$ is a manifold. Let ${\widetilde {P}}:{\mathcal {G}} X=T^1{\widetilde {M}}\rightarrow {\mathbb R}$ be a potential on X, that is, a $\Gamma $ -invariant, boundedFootnote ⁷ and Hölder-continuous real map on $T^1{\widetilde {M}}$ . A potential ${\widetilde {P}}$ is time-reversible if ${\widetilde {P}}={\widetilde {P}}\circ \iota $ , where $\iota $ is the time-reversal map. For all $x,y\in {\widetilde {M}}$ , let us define the amplitude $\int _x^y{\widetilde {P}}$ of ${\widetilde {P}}$ between x and y to be $\int _x^y{\widetilde {P}}=0$ if $x=y$ and otherwise $\int _x^y{\widetilde {P}}= \int _{0}^{d(x,y)} {\widetilde {P}}({\mathtt {g}^{t}} v) \;dt$ where v is the tangent vector at x to the geodesic segment from x to y.

Now assume that X is a tree, with the notation of the beginning of Section 2. Denote by $V{\mathbb X}$ , $E{\mathbb X}$ the sets of vertices and edges of ${\mathbb X}$ , and by ${\overline e}$ the opposite edge of an edge e. Let ${\widetilde {c}}: E{\mathbb X}\rightarrow {\mathbb R}$ be a (logarithmic) system of conductances (see for instance [Reference ZemanianZem]), that is, a $\Gamma $ -invariant, bounded real map on $E{\mathbb X}$ . For every geodesic line $\ell \in {\mathcal {G}} X$ , we denote by $e^+_0(\ell )=\ell ([0,1])\in E{\mathbb X}$ the first edge followed by $\ell $ . We define the potential ${\widetilde {P}}:{\mathcal {G}} X\rightarrow {\mathbb R}$ on X associated with ${\widetilde {c}}$ as the map $\ell \mapsto {\widetilde {c}}\,(e^+_0(\ell ))$ . The potential ${\widetilde {P}}$ is time-reversible if ${\widetilde {c}}(\overline {e})={\widetilde {c}}(e)$ for every $e\in E{\mathbb X}$ . For all $x,y\in V{\mathbb X}$ , let $(e_1,e_2, \dots , e_k)$ be the geodesic edge path in ${\mathbb X}$ between x and y, where $k\in {\mathbb N}$ satisfies $k=0$ if and only if $x=y$ . We define the amplitude of ${\widetilde {P}}$ between x and y to be $\int _x^y{\widetilde {P}}= \sum _{i=1}^{k} \;\;{\widetilde {c}}\,(e_i)$ .

If a potential ${\widetilde {P}}$ is time-reversible, then for all $x,y\in {\widetilde {M}}$ or $x,y\in V{\mathbb X}$ , we have

(13) $$ \begin{align} \int_y^x{\widetilde{P}} =\int_x^y{\widetilde{P}}\circ\iota=\int_x^y{\widetilde{P}}\,. \end{align} $$

We denote by $P:\Gamma \backslash {\mathcal {G}} X\rightarrow {\mathbb R}$ the quotient map of ${\widetilde {P}}$ , that we call a potential on $\Gamma \backslash {\mathcal {G}} X$ .

Recall that every conjugacy class $[\gamma ]$ of loxodromic elements of $\Gamma $ defines a (not necessarily primitive) periodic orbit ${\mathcal {O}}_{[\gamma ]}$ for the geodesic flow $({\mathtt {g}^{t}})_t$ on $\Gamma \backslash {\mathcal {G}} X$ , with (not necessary minimal) length $\lambda ({\mathcal {O}}_{[\gamma ]}) = \lambda (\gamma )$ , so that for every $\ell \in {\mathcal {G}} X$ with $\Gamma \ell \in {\mathcal {O}}_{[\gamma ]}$ , there exists a conjugate $\gamma '$ of $\gamma $ in $\Gamma $ with $\ell ({\mathbb R})= \operatorname {Ax}_{\gamma '}$ and $\ell ^{-1}(\gamma '\ell (0))>0$ . Let ${\mathcal {O}}$ be a periodic orbit for the geodesic flow $({\mathtt {g}^{t}})_t$ on $\Gamma \backslash {\mathcal {G}} X$ . The period of ${\mathcal {O}}$ for the potential ${\widetilde {P}}$ is

$$\begin{align*}\int_{\mathcal{O}} P= \int_{\ell(0)}^{\ell(\lambda({\mathcal{O}}))} \;{\widetilde{P}}\,, \end{align*}$$

where $\ell $ is any element in ${\mathcal {G}} X$ that maps to ${\mathcal {O}}$ (see for instance [Reference Paulin, Pollicott and SchapiraPPS, §3.1] for the independence of the period on the choice of $\ell $ ).

Let I and J be $\Gamma $ -invariant and nonempty subsets of $I_\Gamma $ . A (not necessarily primitive) orbit ${\mathcal {O}}$ of the geodesic flow on $\Gamma \backslash {\mathcal {G}} X$ is called an $\{I,J\}$ -reversible periodic orbit if there exists $[\gamma ]\in {\mathfrak R}_{I,J}$ such that ${\mathcal {O}}={\mathcal {O}}_{[\gamma ]}$ . We denote by ${\mathcal {O}}_{I,J}$ the multiset whose underlying set $\underline {{\mathcal {O}}_{I,J}}$ is the set of $\{I,J\}$ -reversible periodic orbits for the geodesic flow on $\Gamma \backslash {\mathcal {G}} X$ , where the multiplicities are defined by

(14) $$ \begin{align} \forall\;{\mathcal{O}}\in \underline{{\mathcal{O}}_{I,J}},\quad \operatorname{mult}({\mathcal{O}})= \sum_{[\gamma]\in{\mathfrak R}_{I,J}:\,{\mathcal{O}}={\mathcal{O}}_{[\gamma]}}\operatorname{mult}([\gamma])\,. \end{align} $$

The map $\Theta _4: \underline {{\mathfrak R}_{I,J}}\rightarrow \underline {{\mathcal {O}}_{I,J}}$ defined by $[\gamma ]\mapsto {\mathcal {O}}_{[\gamma ]}$ is hence a multiset bijection from ${\mathfrak R}_{I,J}$ to ${\mathcal {O}}_{I,J}$ .

Critical exponent and Gibbs cocycle. Let us fix an arbitrary basepoint $x_*$ in X, which is a vertex of ${\mathbb X}$ when X is a tree. The critical exponent of a potential P is the weighted (by exponential amplitudes) orbital exponential growth rate of the group $\Gamma $ , defined by

$$\begin{align*}\delta_P= \lim_{n\rightarrow+\infty}\;\frac{1}{n}\;\ln\;\Big( \sum_{\gamma\in\Gamma,\;n-1< d(x_*,\gamma x_*)\leq n} \;\; \exp\Big(\int_{x_*}^{\gamma x_*} {\widetilde{P}}\;\Big)\Big)\,. \end{align*}$$

It is independent of the choice of the basepoint $x_*$ . We have $\delta _P\in \;]-\infty ,+\infty [$ since the potential ${\widetilde {P}}$ is bounded, and $\delta _{P\circ \iota }=\delta _P$ .

The (normalised) Gibbs cocycle of the potential ${\widetilde {P}}$ is the function $C:\partial _\infty X\times {\widetilde {M}}\times {\widetilde {M}}\rightarrow {\mathbb R}$ when $X={\widetilde {M}}$ is a manifold or the function $C:\partial _\infty X\times V{\mathbb X}\times V{\mathbb X}\rightarrow {\mathbb R}$ when X is a tree, defined by the following limit of difference of amplitudes for the renormalised potential

$$\begin{align*}(\xi,x,y)\mapsto C_\xi(x,y)= \lim_{t\rightarrow+\infty} \int_y^{\xi_t}({\widetilde{P}}-\delta_P)-\int_{x}^{\xi_t}({\widetilde{P}} -\delta_P)\,, \end{align*}$$

where $t\mapsto \xi _t$ is any geodesic ray in X converging to $\xi $ . The Gibbs cocycle is $\Gamma $ -invariant (for the diagonal action) and locally Hölder-continuous.

Patterson and Gibbs measures. A (normalised) Patterson density for $(\Gamma ,{\widetilde {P}})$ is a $\Gamma $ -equivariant family $(\mu _{x})_{x\in {\widetilde {M}}}$ when X is a manifold, and $(\mu _{x})_{x\in V{\mathbb X}}$ when X is a tree, of pairwise absolutely continuous (positive, Borel) measures on $\partial _\infty X$ , whose support is the limit set $\Lambda \Gamma $ of $\Gamma $ , such that

$$\begin{align*}\gamma_*\mu_x=\mu_{\gamma x}\mathrm{~~~and~~~} \frac{d\mu_x}{d\mu_y}(\xi) = e^{-C_\xi(x,\,y)} \end{align*}$$

for every $\gamma \in \Gamma $ , for all $x,y\in {\widetilde {M}}$ (respectively $x,y\in V{\mathbb X}$ ), and for (almost) every $\xi \in \partial _\infty X$ .

The Hopf parametrisation of ${\mathcal {G}} X$ with basepoint $x_*$ is the Hölder-continuous homeomorphism from ${\mathcal {G}} X$ to $(\partial _\infty X \times \partial _\infty X - \mathrm {Diag})\times R$ , where $R={\mathbb R}$ if X is a manifold and $R={\mathbb Z}$ if X is a tree, defined by $\ell \mapsto (\ell _-,\ell _+,t)$ , where $\ell _-$ , $\ell _+$ are the original and terminal points at infinity of the geodesic line $\ell $ , and t is the algebraic distance along $\ell $ between the footpoint $\ell (0)$ of $\ell $ and the closest point to $x_*$ on the geodesic line $\ell ({\mathbb R})$ .

Assume from now on that ${\widetilde {P}}$ is a time-reversible potential on X. We denote by $dt$ the Lebesgue or counting measure on R. The Gibbs measure on ${\mathcal {G}} X$ associated with the above Patterson density for $(\Gamma ,{\widetilde {P}})$ is the $\sigma $ -finite nonzero measure ${\widetilde {m}}_P$ on ${\mathcal {G}} X$ defined using the Hopf parametrisation by

$$\begin{align*}d{\widetilde{m}}_P(\ell)= e^{C_{\ell_-}(x_*,\,\ell(0))+C_{\ell_+}(x_*,\,\ell(0))}\; d\mu_{x_*}(\ell_-)\;d\mu_{x_*}(\ell_+)\;dt\,. \end{align*}$$

See [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Eq. (4.4)], noting that we assume P to be time-reversible. The measure ${\widetilde {m}}_P$ is independent of the choice of $x_*$ , is $\Gamma $ -invariant and $({\mathtt {g}^{t}})_{t\in R}$ -invariant. Therefore it inducesFootnote ⁸ a $\sigma $ -finite nonzero $({\mathtt {g}^{t}})_{t\in R}\,$ -invariant measure on $\Gamma \backslash {\mathcal {G}} X$ , called the Gibbs measure on $\Gamma \backslash {\mathcal {G}} X$ for the potential P and denoted by $m_P$ .

Skinning measures. Let D be a nonempty, proper, closed and convex subset of X. We denote by $\partial ^1_{\pm } D$ the outer/inner unit normal bundle of $\partial D$ , that is, the space of geodesic rays $\rho :\pm [0,+\infty [\rightarrow\; X$ with point at infinity $\rho _\pm \in \partial _\infty X$ such that $\rho (0)\in \partial D$ , $\rho _\pm \notin \partial _\infty D$ and the closest point projection on D of $\rho _\pm $ is $\rho (0)$ . We consider such a $\rho $ as a generalised geodesic in by requiring it to be constant on $\mp [0,+\infty [\,$ . When X is a manifold and D is a totally geodesic submanifold of X, then the map from the unit normal bundle $\nu ^1 D$ of D to $\partial ^1_{+} D$ , sending a unit normal vector to D to the positive geodesic ray it defines is a homeomorphism.

Using the endpoint homeomorphism $\rho \mapsto \rho _\pm $ from $\partial ^1_{\pm } {D}$ to $\partial _{\infty }X - \partial _{\infty }D$ , we defined in [Reference Parkkonen and PaulinPP1, Reference Parkkonen and PaulinPP3], see also [Reference Broise-Alamichel, Parkkonen and PaulinBrPP], the outer/inner skinning measure Footnote ⁹ ${\widetilde {\sigma }}^\pm _{D}$ of D for $(\Gamma ,P)$ , associated with the above Patterson density, to be the measure on , with support contained in $\partial ^1_{\pm }{D}$ , given for $\rho \in \partial ^1_{\pm }{D}$ by

(15) $$ \begin{align} d{\widetilde{\sigma}}^\pm_D(\rho) = e^{C_{\rho_\pm}(x_*,\,\rho(0))}\;d\mu_{x_*}(\rho_{\pm}) \,. \end{align} $$

Let $I'$ be an index set endowed with a left action of $\Gamma $ . Let ${\mathcal {D}}=(D_i)_{i\in I'}$ be a locally finite $\Gamma $ -equivariant family of nonempty, proper, closed and convex subsets of X. The outer/inner skinning measure of ${\mathcal {D}}$ on is the $\Gamma $ -invariant locally finite measure on defined byFootnote ¹⁰

It induces a locally finite measure on , denoted by $\sigma ^\pm _{{\mathcal {D}}}$ , called the outer/inner skinning measure of ${\mathcal {D}}$ on . Since ${\widetilde {P}}$ is time-reversible, we have $\iota _*\sigma ^\pm _{{\mathcal {D}}}=\sigma ^\mp _{{\mathcal {D}}}$ , and in particular the measures $\sigma ^-_{{\mathcal {D}}}$ and $\sigma ^+_{{\mathcal {D}}}$ have the same total mass:

(16) $$ \begin{align} \|\sigma^-_{{\mathcal{D}}}\|=\|\sigma^+_{{\mathcal{D}}}\|\,. \end{align} $$

5 Counting strongly reversible closed geodesics

In this Section, let $(X,\Gamma )$ be as in the beginning of Section 2. Let $x_*$ be an arbitrary basepoint in X, that we take to be a vertex of ${\mathbb X}$ when X is a tree.

Let I and J be two nonempty and $\Gamma $ -invariant subsets of $I_\Gamma $ . Let ${\mathcal {F}}_I= (F_\alpha )_{\alpha \in I}$ (respectively ${\mathcal {F}}_J= (F_\alpha ) _{\alpha \in J}$ ) be the $\Gamma $ -equivariant family of the fixed point sets of the elements of I (respectively J). Let $\sigma ^+_{I} = \sigma ^+_{{\mathcal {F}}_I}$ be the outer skinning measure of ${\mathcal {F}}_I$ , and $\sigma ^-_{J} = \sigma ^-_{{\mathcal {F}}_J}$ the inner skinning measure of ${\mathcal {F}}_J$ . Let ${\widetilde {P}}:{\mathcal {G}} X\rightarrow {\mathbb R}$ be a time-reversible potential with positive critical exponent $\delta _P$ . When X is a tree, we furthermore assume that the smallest nonempty and $\Gamma $ -invariant simplicial subtree ${\mathbb X}'$ of ${\mathbb X}$ is uniform,Footnote ¹¹ without vertices of degree $2$ .

When X is a tree, we denote by $L_\Gamma $ the length spectrum of $\Gamma $ , that is, the subgroup of ${\mathbb Z}$ generated by the translation lengths in X of the loxodromic elements of $\Gamma $ . By [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Lem. 4.18], under the assumption of the finiteness of the measure $m_P$ , the above assumption on the tree ${\mathbb X}$ implies that $L_\Gamma $ is either ${\mathbb Z}$ or $2{\mathbb Z}$ . When $L_\Gamma ={\mathbb Z}$ , the geodesic flow $({\mathtt {g}^{t}})_{t\in {\mathbb Z}}$ is mixing for the Gibbs measure $m_{P}$ on $\Gamma \backslash {\mathcal {G}} X$ by [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 4.17]. Let be the subset of elements such that ${\displaystyle \lim _{t\rightarrow -\infty }}\,\ell (t)$ , $\ell (0)$ , ${\displaystyle \lim _{t\rightarrow +\infty }}\,\ell (t)$ are at even distance of $x_*$ or belong to $\partial _\infty X$ . Let

By [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 4.17], when $L_\Gamma =2{\mathbb Z}$ , the subsets and ${\mathcal {G}}_{\mathrm {even}} X$ are $\Gamma $ -invariant, and the even times geodesic flow $({\mathtt {g}^{2t}})_{t\in {\mathbb Z}}$ is mixing for the restriction $m_{P,\, \mathrm {even}}$ to $\Gamma \backslash {\mathcal {G}}_{\mathrm {even}} X$ of the Gibbs measure $m_{P}$ . We also denote by $\sigma ^+_{I,\,\mathrm {even}}$ and $\sigma ^-_{J,\,\mathrm {even}}$ the restriction of $\sigma ^+_I$ and $\sigma ^+_J$ to .

Let $\delta _P^*= \delta _P$ when X is a manifold. When X is a tree, let $\delta _P^*=1- e^{-\delta _P}$ when $L_\Gamma ={\mathbb Z}$ and $\delta _P^*= 1-e^{-2\delta _P}$ when $L_\Gamma =2{\mathbb Z}$ . For every $s>0$ when X is a manifold, and every $s\in {\mathbb N}-\{0\}$ when X is a tree with $L_\Gamma ={\mathbb Z}$ , let $\operatorname {Perp}^*({\mathcal {F}}_I,{\mathcal {F}}_J,s) = \operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J,s)$ , $IJ^*(s)=IJ(s)$ and ${\mathfrak R} ^*_{I,J} (s)={\mathfrak R}_{I,J} (s)$ .

When X is a tree with $L_\Gamma =2{\mathbb Z}$ , for every $s\in 2{\mathbb N}-\{0\}$ , let $\operatorname {Perp}^*({\mathcal {F}}_I,{\mathcal {F}}_J,s)$ be the multiset restriction of the common perpendiculars in $\operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J,s)$ with both endpoints (which are vertices of ${\mathbb X}$ ) at even distance from the basepoint $x_*$ . Note that when the fixed point sets of the elements in $I\cup J$ are at even distance from the base point $x_*$ , by the relations between the three distances between three points in a tree, we have $\operatorname {Perp}^*({\mathcal {F}}_I,{\mathcal {F}}_J,s) = \operatorname {Perp}({\mathcal {F}}_I, {\mathcal {F}}_J,s)$ . Let also $IJ^*(s)$ be the multiset restriction of $IJ(s)$ consisting of its elements $\Gamma (\alpha ,\beta )$ with both endpoints of the common perpendicular ${\widetilde {\rho }}_{\alpha ,\beta }$ at even distance from $x_*$ . Finally let ${\mathfrak R}^*_{I,J}(2s)$ be the image in ${\mathfrak R}_{I,J} (2s)$ of $IJ^*(s)$ by the multiset bijection $\Theta _2$ defined in Proposition 3.3. Note that the multiset bijection $\Theta _1$ defined in that proposition sends $IJ^*(s)$ to $\operatorname {Perp}^*({\mathcal {F}}_I,{\mathcal {F}}_J,s)$ .

Recall the following standard convention on sums over multisets. Let $S=(\underline S,\operatorname {mult})$ be a finite real-valued multiset. For every map f from $\underline {S}$ to a real vector space, let

$$ \begin{align*}\sum_{a\in S} f(a)=\sum_{a\in\underline{S}}\; \operatorname{mult}(a)\,f(a)\,. \end{align*} $$

For $T\in{\mathbb R}$ if X is a manifold, $T\in 2{\mathbb Z}, 4{\mathbb Z}$ if X is a tree with $L_{\gamma}={\mathbb Z}, 2{\mathbb Z}$ respectively, let

(17) $$ \begin{align} {\mathcal{N}}_{I,J,P}(T)=\sum_{[\gamma]\,\in \,{\mathfrak R}^*_{I,J}(T)} \;e^{\frac{1}{2}\int_{[\gamma]}P} \end{align} $$

be the counting function of conjugacy classes of $\{I,J\}$ -reversible loxodromic elements of $\Gamma $ with translation length at most T, with multiplicities given by Equation (6) and with weights given by their exponential half-periods.

When $X={\widetilde {M}}$ is a manifold, the mixing assumption for the error term in Theorem 5.1 (1) holds for instance if ${\widetilde {M}}$ is a real hyperbolic space ${\mathbb H}^n_{\mathbb R}$ and $\Gamma $ is geometrically finite by [Reference Li and PanLP], or if $\Gamma $ is an arithmetic lattice in the isometry group of a negatively curved symmetric space ${\widetilde {M}}$ by the works of Clozel, Kleinbock and Margulis, as explained in [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, §9.1]. When X is a tree, the error term in Theorem 5.1 (1) (respectively (2)) holds if X is the Bruhat-Tits tree of a rank one simple algebraic group G over a nonarchimedean local field and $\Gamma $ is any lattice in G with $L_\Gamma ={\mathbb Z}$ (respectively with $L_\Gamma =2{\mathbb Z}$ ) by [Reference LubotzkyLub, Theo. C].

Theorem 1.1 (1) in the introduction follows from Theorem 5.1 (1) by restricting to the manifold case and by taking ${\widetilde {P}}=0$ (which is time-reversible), in which case $\delta _P=\delta _\Gamma>0$ and $m_P=m_{\mathrm {BM}}$ , since ${\mathcal {N}}_{I,J}(T)$ as defined in the introduction is equal to ${\mathcal {N}}_{I,J,0}(T)$ as defined in Equation (17).

Proof. The main idea of the proof is to relate the counting function of strongly reversible closed geodesics in $\Gamma \backslash X$ with the counting function of the common perpendiculars between the pairs of fixed point sets of the involutions in I and J, using the preliminary work of Section 2.

Let $(\alpha ,\beta ) \in {\widetilde {IJ}}$ . Let $\gamma =\beta \alpha $ and let $\rho _{\alpha ,\beta }$ be the $\Gamma $ -orbit of the common perpendicular $[x,y]$ from $F_\alpha $ to $F_\beta $ with $x\in F_\alpha $ , so that $\beta y=y$ . Let us define $\int _{\rho _{\alpha ,\beta }} P=\int _x^y{\widetilde {P}}$ , which is well defined since ${\widetilde {P}}$ is $\Gamma $ -invariant. Since ${\widetilde {P}}$ is time-reversible, we have $\int _x^y{\widetilde {P}}= \int _y^{x}{\widetilde {P}}\circ \iota = \int _y^{x}{\widetilde {P}}$ . Hence

(20) $$ \begin{align} \int_{\rho_{\alpha,\beta}} P=\int_{\rho_{\beta,\alpha}} P\,. \end{align} $$

Since ${\widetilde {P}}$ is $\Gamma $ -invariant, we have $\int _x^y{\widetilde {P}}=\int _y^x {\widetilde {P}}=\int _{\beta y}^{\beta x}{\widetilde {P}}= \int _y^{\beta x}{\widetilde {P}}$ . Hence since x belongs to the translation axis of $\gamma $ by Lemma 2.3, since y is the midpoint of $[x,\gamma x]= [x,\beta x]$ , and by the additivity properties of the amplitudes, we have

(21) $$ \begin{align} \int_{[\gamma]} P=\int_x^{\gamma x}{\widetilde{P}}=\int_x^{y}{\widetilde{P}}+\int_y^{\beta x}{\widetilde{P}} =2\int_{\rho_{\alpha,\beta}} P\,. \end{align} $$

The assumptions of [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Coro. 12.3] when X is a manifold, and of [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.9] when X is a tree with $L_\Gamma ={\mathbb Z}$ , are satisfied by the assumptions of Theorem 5.1 (1). The assumptions of [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.12] when X is a tree with $L_\Gamma =2{\mathbb Z}$ are satisfied by the assumptions of Theorem 5.1 (2) and by the second part of [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 4.17] for the required mixing property. We claim that the counting parts in these results, with the above definitions of $\delta _P^*$ and $\operatorname {Perp}^*$ , imply that

(22)

as $T\rightarrow +\infty $ with $T\in {\mathbb R}$ if X is a manifold and $T\in 2{\mathbb Z}$ if X is a tree with $L_\Gamma ={\mathbb Z}$ , and that

(23)

as $T\rightarrow +\infty $ with $T\in 4{\mathbb Z}$ if X is a tree with $L_\Gamma =2{\mathbb Z}$ .

The conclusion of [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Coro. 12.3] is formulated in a way that is not directly applicable, considering the difference between the counting function ${\mathcal {N}}_{{\mathcal {F}}_I,{\mathcal {F}}_J,P}$ in [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, page 255] and the above counting function $\sum _{\rho \,\in \, \operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J, \frac {T}{2})} \;e^{\int _{\rho }P}$ . The conclusions of [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.9 and Theo. 12.12] would only apply when $\Gamma \backslash I$ and $\Gamma \backslash J$ are reduced to one element (that is, when I and J contain only one conjugacy class of involutions, as for Corollary 1.3). Hence we need to go over the proofs of these results and explain the work on the multiplicities in order to obtain Equations (22) and (23), without stating in full these statements and their proofs.

The process is the following one. The counting parts in these three statements [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Coro. 12.3, Theo. 12.9 and Theo. 12.12] are deduced from three claims of narrow convergence of measures in the quotient space by integrating on the constant function $1$ . These three claims are themselves consequences of three convergence statements of measures on the product space stated as [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 11.1, Theo. 11.9 and Eq. (11.28)]. These last three convergence statements have the form

(24) $$ \begin{align} \lim_{t\rightarrow+\infty}\sum_{a\in {\mathcal{A}}_t} \;{\widetilde{\nu}}_a\;=\;{\widetilde{\nu}}_\infty\,, \end{align} $$

where

• • for every $t>0$ , the index set ${\mathcal {A}}_t$ is endowed with a left action of $\Gamma \times \Gamma $ ,

• • the measures ${\widetilde {\nu }}_a$ for $a\in {\mathcal {A}}_t$ are positive multiples of Dirac masses on and the map $a\mapsto {\widetilde {\nu }}_a$ is equivariant for the action of $\Gamma \times \Gamma $ .

In order to be continuous, the process of defining measures on the quotient by branched coverings (see for instance [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, §2.6]) consists of the following operations. The new index set of the sum is the quotient modulo the action of the group $\Gamma \times \Gamma $ on the index set ${\mathcal {A}}_t$ . The image measure $\nu _a$ of ${\widetilde {\nu }}_a$ in is divided by the cardinality of the stabiliser in $\Gamma \times \Gamma $ of the support of ${\widetilde {\nu }}_a$ . Let $\nu _\infty $ be the measure on induced by ${\widetilde {\nu }}_\infty $ . This gives a convergence, which turns out to be a narrow convergence as explained in [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Coro. 12.3, Theo. 12.9 and Theo. 12.12], of the form

(25) $$ \begin{align} \lim_{t\rightarrow+\infty}\sum_{a\in (\Gamma\times\Gamma)\backslash {\mathcal{A}}_t}\; \frac{1}{\operatorname{Card} (\operatorname{Stab}_{\Gamma\times\Gamma}(a))} \;\nu_a =\nu_\infty\,. \end{align} $$

Let us now explain how we apply this process. For every $\alpha \in I$ (resp. $\beta \in J$ ), let ${\overline \alpha }$ (resp. ${\overline \beta }$ ) be its class in (resp. ). When X is a manifold, the statement in [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 11.1] (taking into account that the action of $\Gamma $ on I is by conjugation) uses the index set

where t varies in $]0,+\infty [\,$ . The action of $\Gamma \times \Gamma $ on ${\mathcal {A}}_t$ is given by

$$\begin{align*}(\gamma_1,\gamma_2)\cdot ({\overline \alpha},{\overline \beta},\gamma) =(\gamma_1\,{\overline \alpha}\,\gamma_1^{-1}, \gamma_2\, {\overline \beta}\, \gamma_2^{-1},\gamma_1\,\gamma\,\gamma_2^{-1})\,. \end{align*}$$

If $(\gamma _1,\gamma _2)\in \Gamma \times \Gamma $ fixes $({\overline \alpha },{\overline \beta },\gamma )\in {\mathcal {A}}_t$ , then $\gamma _2= \gamma ^{-1} \gamma _1\gamma $ is uniquely determined by $\gamma _1$ . Note that $\gamma _2=\gamma ^{-1} \gamma _1\gamma $ belongs to $\Gamma _{F_\beta }$ if and only if $\gamma _1$ belongs to $\gamma \Gamma _{F_\beta } \gamma ^{-1}= \Gamma _{\gamma F_\beta } =\Gamma _{F_{\gamma \beta \gamma ^{-1}}}$ . For every $(\alpha _0,\gamma _0)\in I\times \Gamma $ , by the definition of the equivalence relation in Formula (3), we have if and only if $F_{\alpha _0} = F_{\gamma _0\,\alpha _0\,\gamma _0^{-1}}$ , that is, if and only if $\gamma _0$ belongs to the stabiliser $\Gamma _{F_{\alpha _0}}$ of $F_{\alpha _0}$ in $\Gamma $ . Applying this for both $(\alpha _0,\gamma _0) =(\alpha ,\gamma _1)$ and $(\alpha _0,\gamma _0)=(\beta ,\gamma _2)$ with I replaced by J, this proves that the stabiliser of $({\overline \alpha }, {\overline \beta },\gamma )\in {\mathcal {A}}_t$ in $\Gamma \times \Gamma $ is isomorphic to $\Gamma _{F_\alpha }\cap \Gamma _{F_{\gamma \beta \gamma ^{-1}}}$ , by the map $(\gamma _1, \gamma _2) \mapsto \gamma _1$ . Hence, these two finite groups have the same cardinality.

For every $t\in [0,+\infty [\,$ , let us consider the set

endowed with the diagonal action $\gamma \cdot ({\overline \alpha },{\overline \beta })= (\gamma \,{\overline \alpha }\,\gamma ^{-1},\gamma \,{\overline \beta }\,\gamma ^{-1})$ by conjugation of $\Gamma $ . The map ${\widetilde {\Theta }}: ({\overline \alpha },{\overline \beta })\mapsto ({\overline \alpha }, {\overline \beta },e)$ from the set ${\mathcal {A}}^{\prime }_t$ endowed with its action of $\Gamma $ to the set ${\mathcal {A}}_t$ endowed with its action of $\Gamma \times \Gamma $ is equivariant for the diagonal group morphism from $\Gamma $ into $\Gamma \times \Gamma $ . It hence induces, by taking quotients, a map

$$\begin{align*}\Theta:\Gamma \backslash {\mathcal{A}}^{\prime}_t\rightarrow(\Gamma\times\Gamma)\backslash {\mathcal{A}}_t\,. \end{align*}$$

Note that for all $\alpha \in I$ , $\beta \in J$ and $\gamma \in \Gamma $ , we have $(e,\gamma ) \cdot ({\overline \alpha }, {\overline \beta },\gamma )= ({\overline \alpha }, \gamma \,{\overline \beta }\, \gamma ^{-1},e)$ . Hence, every element of ${\mathcal {A}}_t$ is in the same $(\Gamma \times \Gamma )$ -orbit as an element of the image of ${\widetilde {\Theta }}$ , and the map $\Theta $ is surjective. The map $\Theta $ is also injective, since for all $({\overline \alpha },{\overline \beta }), ({\overline \alpha '}, {\overline \beta '})\in {\mathcal {A}}^{\prime }_t$ , if there exists $(\gamma _1,\gamma _2)\in \Gamma \times \Gamma $ such that $(\gamma _1, \gamma _2) \cdot ({\overline \alpha },{\overline \beta },e) = ({\overline \alpha '}, {\overline \beta '},e)$ , then $\gamma _1=\gamma _2$ , and $({\overline \alpha },{\overline \beta })$ and $({\overline \alpha '}, {\overline \beta '})$ are in the same $\Gamma $ -orbit in ${\mathcal {A}}^{\prime }_t$ .

The map $\Theta ':\Gamma (\alpha ,\beta )\mapsto \Gamma ({\overline \alpha },{\overline \beta })$ from the set $\underline {IJ(t)}$ (defined in Equation (5)) to the set $\Gamma \backslash {\mathcal {A}}^{\prime }_t$ is surjective. Its fiber over $\Gamma ({\overline \alpha }, {\overline \beta })$ has cardinality $|\Gamma _{F_\alpha }/Z_\Gamma (\alpha )|\; |\Gamma _{F_\beta }/ Z_\Gamma (\beta )|$ by Lemma 2.6. Hence

• • by the definitions of a sum over a multiset and of the multiplicity of a common perpendicular between elements of ${\mathcal {F}}_I$ and ${\mathcal {F}}_J$ given in Equation (9) for the first equality below,

• • by the orders of the fibers of $\Theta ':\underline {IJ(t)} \rightarrow (\Gamma \backslash {\mathcal {A}}^{\prime }_t)$ for the second equality below,

• • using the bijection $\Theta :(\Gamma \backslash {\mathcal {A}}^{\prime }_t)\rightarrow (\Gamma \times \Gamma ) \backslash {\mathcal {A}}_t$ for the third equality below,

we have

$$ \begin{align*} \sum_{\rho\,\in\, \operatorname{Perp}({\mathcal{F}}_I,\,{\mathcal{F}}_J,\,t)} \;e^{\int_{\rho}P}&= \sum_{\Gamma(\alpha,\beta)\,\in\, \underline{IJ(t)}} \frac{e^{\int_{\rho_{\alpha,\beta}}P}}{{\operatorname{Card}}(\Gamma_{F_\alpha}\cap\Gamma_{F_\beta})\; |\Gamma_{F_\alpha}/Z_\Gamma(\alpha)|\; |\Gamma_{F_\beta}/Z_\Gamma(\beta)|} \\&=\sum_{\Gamma\cdot({\overline \alpha},{\overline \beta})\,\in\, \Gamma \backslash {\mathcal{A}}^{\prime}_t} \frac{e^{\int_{\rho_{\alpha,\beta}}P}}{{\operatorname{Card}}(\Gamma_{F_\alpha}\cap\Gamma_{F_\beta})} \\&=\sum_{(\Gamma\times\Gamma)\cdot({\overline \alpha},{\overline \beta},\gamma)\,\in\, (\Gamma\times\Gamma) \backslash {\mathcal{A}}_t} \frac{e^{\int_{\rho_{\alpha,\gamma\beta\gamma^{-1}}}P}} {{\operatorname{Card}}(\Gamma_{F_\alpha}\cap\Gamma_{F_{\gamma\beta\gamma^{-1}}})}\,. \end{align*} $$

Therefore, taking $t=\frac {T}{2}$ , Equation (22) when X is a manifold (so that $\delta _P^*= \delta _P$ ) then follows from [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 11.1] applied with ${\mathcal {D}}^-={\mathcal {F}}_I$ and ${\mathcal {D}}^+={\mathcal {F}}_J$ , by the process explained above starting from Equation (24), passing through Equation (25) and integrating on the constant function $1$ .

When X is a tree and $L_\Gamma ={\mathbb Z}$ , the proof of Equation (22) is similar, taking $t=\frac {T}{2}\in {\mathbb N}$ (hence $T\in 2{\mathbb N}$ ), replacing [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 11.1] by [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 11.9] applied with ${\mathcal {D}}^-={\mathcal {F}}_I$ and ${\mathcal {D}}^+={\mathcal {F}}_J$ , using the above definition of $\delta _P^*=1-e^{-\delta _P}$ and the definition of the amplitudes for trees given in Section 4.

When X is a tree and $L_\Gamma =2{\mathbb Z}$ , the proof of Equation (23) is also similar, taking $t=\frac {T}{2}\in 2{\mathbb N}$ (hence $T\in 4{\mathbb N}$ ), replacing [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 11.1] by [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Eq. (11.28)] applied with ${\mathcal {D}}^-={\mathcal {F}}_I$ and ${\mathcal {D}}^+={\mathcal {F}}_J$ , using the above definitions of $\delta _P^*=1-e^{-2\delta _P}$ and $\operatorname {Perp}^*$ , noting that we have $m_{P,\, \mathrm {even}}= \frac {\|m_P\|}{2}$ .

Let us prove Assertion (1) of Theorem 5.1. Under its assumptions, using respectively in the following sequence of equalities and equivalences

• • the definition of ${\mathcal {N}}_{I,J,P}(T)$ in Equation (17) for the first equality,

• • the fact that the map $\Theta _2$ in Proposition 3.3 is a multiset bijection sending $IJ(\frac {T}{2})$ to ${\mathfrak R}_{I,J}(T)$ , and Equation (21), for the second equality,

• • the fact that the map $\Theta _1$ in Proposition 3.3 is a multiset bijection sending $IJ(\frac {T}{2})$ to $\operatorname {Perp}({\mathcal {F}}_I,{\mathcal {F}}_J,\frac {T}{2})$ , for the third equality, and

• • Equation (22) for the final equivalence,

as $T\rightarrow +\infty $ with $T\in 2{\mathbb Z}$ if X is a tree with $L_\Gamma ={\mathbb Z}$ , we have

(26)

This proves Equation (18).

When $X={\widetilde {M}}$ is a manifold, the error term claim of Theorem 5.1 (1) follows from the analogous error term in Equation (22), obtained in [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.7 (2)] using

• • the fact that when ${\widetilde {P}}=0$ , ${\widetilde {M}}$ is locally symmetric and $\Gamma \backslash {\widetilde {M}}$ has finite volume, then $m_P=m_{\mathrm {BM}}$ is the (smooth) Liouville measure, which is finite,

• • the fact that the fixed point sets of involutions of $\Gamma $ are (smooth) totally geodesic submanifolds of ${\widetilde {M}}$ ,

• • the exponential mixing assumption that is needed in [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.7 (2)].

When X is a tree with $L_\Gamma ={\mathbb Z}$ , the error term claim of Theorem 5.1 (1) follows from the analogous error term in Equation (22), now obtained by [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.16], and more precisely in its following Remark (i) page 281.

The proof of Assertion (2) of Theorem 5.1 when X is a tree with $L_\Gamma =2{\mathbb Z}$ is similar. Note that by the above definitions of $IJ^*(T)$ , $\operatorname {Perp}^*({\mathcal {F}}_I,{\mathcal {F}}_J,T)$ and ${\mathfrak R}^*_{I,J}(T)$ , and as mentioned before Equation (17), the map $\Theta _1$ sends $IJ^*(T)$ to $\operatorname {Perp}^*({\mathcal {F}}_I,{\mathcal {F}}_J, T)$ and $\Theta _2$ sends $IJ^*(T)$ to ${\mathfrak R}^*_{I,J}(2T)$ . We then obtain Equation (26) when $T\in 4{\mathbb Z}$ by replacing Equation (22) by Equation (23). For the error term, we replace Remark (i) following [Reference Broise-Alamichel, Parkkonen and PaulinBrPP, Theo. 12.16] by its following Remark (ii).