2.1 Graphs of groups and spaces

Graphs of groups were introduced as combinatorial objects in [Reference SerreSer77]. In [Reference Scott and WallSW79], Scott and Wall introduced graphs of spaces in order to study graphs of groups topologically. Since we will use both viewpoints in this paper, we take the time to introduce them here.

Throughout this subsection, $\Gamma$ denotes a directed graph, and $\mathsf{Vert}(\Gamma)$ and $\mathsf{Edge}(\Gamma)$ denote the vertex and edge sets of $\Gamma$ , respectively. For any edge $e \in \mathsf{Edge}(\Gamma)$ , let $\mathsf{o}(e) \in \mathsf{Vert}(\Gamma)$ and $\mathsf{t}(e) \in \mathsf{Vert}(\Gamma)$ denote the origin and terminus of e.

1. (1) a connected directed graph $\Gamma$ , called the underlying graph of $\mathcal G$ ;

2. (2) groups $G_v$ and $G_e$ for every vertex $v \in \mathsf{Vert}(\Gamma)$ and edge $e \in \mathsf{Edge}(\Gamma)$ ;

3. (3) monomorphisms $\varphi_{e,\mathsf{o}} \colon G_e \longrightarrow G_{\mathsf{o}(e)}$ and $\varphi_{e,\mathsf{t}} \colon G_e \longrightarrow G_{\mathsf{t}(e)}$ for every edge $e \in \mathsf{Edge}(\gamma)$ .

The groups $G_v$ and $G_e$ are called the vertex groups and edge groups of $\mathcal G$ . The monomorphisms $\varphi_{e,\mathsf{o}}$ and $\varphi_{e,\mathsf{t}}$ are called the edge maps of $\mathcal G$ .

We now review two ways to look at the fundamental group of a graph of groups.

Definition 2.2. (Based fundamental group). With the same notation as in Definition 2.1, let $v_0 \in \mathsf{Vert}(\Gamma)$ be a base vertex and for each $e \in \mathsf{Edge}(\gamma)$ introduce the formal symbol $t_e$ . Let $P(\mathcal G)$ be the group freely generated by the vertex groups $G_v$ and the symbols $t_e$ subject to the relations $t_e \varphi_{e,\mathsf{t}}(g) t_e^{-1} = \varphi_{e,\mathsf{o}}(g)$ for $e \in \mathsf{Edge}(\Gamma)$ and $g \in G_e$ . The fundamental group of the graph of groups based at $v_0$ , denoted by $\pi_1(\mathcal G, v_0)$ , is the subgroup of $P(\mathcal G)$ consisting of the elements that can be represented as words $g_0 t_{e_1}^{\varepsilon_1} g_1 \cdots t_{e_n}^{\varepsilon_n} g_n$ , where $g_i \in G_{\mathsf{t}(e_i)}$ , where

\[ \varepsilon_i = \pm 1, \quad \begin{cases} g_i \in F_{\mathsf{t}(e_i)} & \text{if} \ \varepsilon_i = 1 \\ g_i \in F_{\mathsf{o}(e_i)} & \text{if} \ \varepsilon_i = -1, \end{cases} \quad g_0, g_n \in G_{v_0}, \]

and where $(e_1, \ldots, e_n)$ forms a (not necessarily directed) loop.

An equivalent way to look at the fundamental group is explained in the following definition. This will be used when considering splittings of $\mathcal G$ over simpler subgraphs of groups as in Proposition 3.9.

A graph of groups is finite if its underlying graph is finite. If G is isomorphic to the fundamental group of a graph of groups, we say that G splits as a graph of groups. In this situation, we will often abuse terminology and say that G is a graph of groups. If the vertex and edge groups of a graph of groups $\mathcal G$ lie in classes $\mathcal C$ and $\mathcal D$ , respectively, then we will say that $\mathcal G$ (or its fundamental group) is a graph of $\mathcal C$ groups with $\mathcal D$ edge groups. We will be mostly interested in graphs of free groups with cyclic edge groups in this paper. A notable subclass which will appear is the class of generalised Baumslag–Solitar groups, which are the groups that split as finite graphs of $\mathbb{Z}$ s with $\mathbb{Z}$ edge groups.

1. (1) there is an injection $\Upsilon \hookrightarrow \Gamma$ (via which we think of $\Upsilon$ as a subgraph of $\Gamma$ );

2. (2) there are inclusions $f_v \colon H_v \hookrightarrow G_v$ and $f_e \colon H_e \hookrightarrow G_e$ for all vertices and edges of $\Upsilon$ (via which we think of every $H_v$ (respectively, $H_e$ ) as a subgroup of $G_v$ (respectively, $G_e$ ));

3. (3) $H_e = H_{\mathsf{o}(e)} \cap G_e$ and $H_e = H_{\mathsf{t}(e)} \cap G_e$ for every $e \in \mathsf{Edge}(\Upsilon)$ ; and

4. (4) the diagrams

commute for all $e \in \mathsf{Edge}(\Upsilon)$ , where the horizontal maps are the edge maps of the respective graphs of groups.

Finally, note that if H is an arbitrary subgroup of a graph of groups G, then H inherits a graph of groups structure, which comes from the action of H on the Bass–Serre tree associated to G (see [Reference SerreSer77, Théorème 13, Chapitre I, §5]).

We will often switch between the graph of groups and graph of spaces viewpoint, the latter of which we introduce now.

1. (1) a connected directed graph $\Gamma$ , called the underlying graph of $\mathcal X$ ;

2. (2) based connected CW complexes $(X_v,x_v)$ and $(X_e,x_e)$ for every vertex $v \in \mathsf{Vert}(\Gamma)$ and edge $e \in \mathsf{Edge}(\Gamma)$ ;

3. (3) based $\pi_1$ -injective continuous maps $f_{e,\mathsf{o}} \colon X_e \longrightarrow X_{\mathsf{o}(e)}$ and $f_{e,\mathsf{t}} \colon X_e \longrightarrow X_{\mathsf{t}(e)}$ for every edge $e \in \mathsf{Edge}(\gamma)$ .

The spaces $X_v$ and $X_e$ are called the vertex spaces and the edge spaces of $\mathcal X$ . The maps $f_{e,\mathsf{o}}$ and $f_{e,\mathsf{t}}$ are called the edge maps. The geometric realisation of $\mathcal X$ is the quotient of the space

\[ X = \bigg(\bigsqcup_{v \in \mathsf{Vert}(\Gamma)} X_v\bigg) \sqcup \bigg( \bigsqcup_{e \in \mathsf{Edge}(\Gamma)} X_e \times [0,1] \bigg), \]

by the relations $f_{e,\mathsf{o}}(x) \sim (x,0)$ and $\varphi_{e,1}(x) \sim (x,1)$ for all $x \in X_e$ and all $e \in \mathsf{Vert}(E)$ . The fundamental group of the topological space $\mathcal X$ based at $x_{v_0}$ , denoted by $\pi_1(\mathcal X,x_{v_0})$ , is defined to be $\pi_1(X, x_{v_0})$ . When no confusion arises, we will usually refer to the geometric realisation of $\mathcal X$ as a graph of spaces.

There is a correspondence between graphs of spaces and graphs of groups. If $\mathcal G$ is a graph of groups, then a graph of spaces $\mathcal X$ can be constructed as follows. For each $v \in \mathsf{Vert}(\Gamma)$ and $e \in \mathsf{Edge}(\Gamma)$ , let $X_v = K(G_v,1)$ and $X_e = K(G_e,1)$ , and let $f_{e,\mathsf{o}}$ and $f_{e,\mathsf{t}}$ be maps inducing $\varphi_{e,\mathsf{o}}$ and $\varphi_{e,\mathsf{t}}$ . Then there is an isomorphism between $\pi_1(\mathcal G, v_0)$ and $\pi_1(\mathcal X, x_{v_0})$ (which depends on the choices of $v_0$ and $x_{v_0}$ up to conjugation). Similarly, given a graph of spaces, we can form a graph of groups with vertex groups $\pi_1(X_v, x_v)$ , edge groups $\pi_1(X_e, x_e)$ , and edge maps $(f_{e,\mathsf{o}})_*$ and $(f_{e,\mathsf{t}})_*$ .

One of our main results concerns graphs of groups where all the vertex groups are free groups and all the edge groups are infinite cyclic. Any such group will be referred to as a graph of free groups with cyclic edge groups. Such groups are realised as the fundamental group of a graph of graphs with $S^1$ edge spaces, by which we understand a graph of spaces where every vertex space is a graph and every edge space is a copy of the circle $S^1$ .

The notion of a precovering will appear throughout § 4 and 5, so we recall it here. It shows up naturally when completing a compact subspace of a covering space to a finite-sheeted covering.

commute. The domain X’ is called a precover.

A precovering $X'\longrightarrow X$ is a covering if and only if all the elevations of edge maps of X to X’ are edge maps of X’. The fact that covering maps induce injections on fundamental groups also applies to precoverings.

We will also require subgraphs of spaces, which induce subgraphs of groups in the sense of Definition 2.4.

1. (1) there is an injection $\Upsilon \hookrightarrow \Gamma$ (via which we think of $\Upsilon$ as a subgraph of $\Gamma$ );

2. (2) there are $\pi_1$ -injective inclusions $f_v \colon Y_v \hookrightarrow X_v$ and $f_e \colon Y_e \hookrightarrow X_e$ for all edges and vertices of $\Upsilon$ (via which we think of every $Y_v$ (respectively, $Y_e$ ) as a subspace of $X_v$ (respectively, $X_e$ ));

3. (3) $\pi_1(Y_e) = \pi_1(Y_{\mathsf{o}(e)}) \cap \pi_1(X_e)$ and $\pi_1(Y_e) = \pi_1(Y_{\mathsf{t}(e)}) \cap \pi_1(X_e)$ for every edge $e \in \mathsf{Edge}(\Upsilon)$ ; and

4. (4) the diagrams

commute for all $e \in \mathsf{Edge}(\Upsilon)$ , where the horizontal maps are edge maps in the corresponding graphs of spaces.

We close by remarking that if X decomposes as a graph of spaces and $Y \longrightarrow X$ is a covering space, then Y inherits a graph of spaces structure where every vertex (respectively, edge) space of Y covers some vertex (respectively, edge) space of X.

2.2 Homology of groups

Unless stated otherwise, all modules are assumed to be left modules. Let R be a ring. Given a right R-module M and a left R-module N, we can define the abelian group ${\rm Tor}_i^R(M, N)$ . By definition, ${\rm Tor}_0^R(M, N)\cong M\otimes_R N$ as an abelian group. In general, the functors ${\rm Tor}_n^R(M,-)$ are the derived functors of $M \otimes_R -$ . More concretely, we choose a projective resolution $P_\bullet \longrightarrow N \longrightarrow 0$ and define ${\rm Tor}_n^M(M,N) := H_n(M \otimes_R P_\bullet)$ .

Let S be another ring. If M is additionally an (S, R)-bimodule, then ${\rm Tor}_n^R(M, N)$ is naturally a left S-module for all n. Similarly, if N is an (R, S)-bimodule, then ${\rm Tor}_n^R(M, N)$ is naturally a right S-module. A standard tool we will use is the long exact sequence in Tor associated to a short exact sequence of modules. Let $0 \longrightarrow N_1 \longrightarrow N_2 \longrightarrow N_3 \longrightarrow 0$ be a short exact sequence of R-modules and let M be an (S, R)-bimodule. Then there is a long exact sequence of left S-modules of the form

A standard reference for this material is [Reference WeibelWei94, Chapters 2 and 3].

Let G be a group and let M be an R[G]-module. As in [Reference BrownBro94, Chapter III, Section 2], the n-dimensional homology of G with coefficients in M is given by

\[ H_n(G;M) := {\rm Tor}_n^{R[G]}(R,M),\]

where R denotes the trivial right R[G]-module. Chiswell’s Mayer–Vietoris exact sequence will be a very useful tools when establishing the $L^2$ -Hall property for certain graphs of groups.

Given a field K and a group G, denote by $I_G$ the augmentation ideal of the group ring K[G] (in practice, this notation will present no ambiguity as the choice of coefficient field K will be clear from the context). Given a subgroup $H \leqslant G$ , we denote by $I_H^G$ the left K[G]-submodule of $I_G$ generated by $I_H$ . In addition, even if H is not normal in G, we will write $K[G/H]$ to refer to the left K[G]-module of left cosets of H in G. The following canonical isomorphisms will be useful later.

1. (1) The canonical map $K[G]\otimes_{K[H]}I_H\longrightarrow I_H^G$ that sends $a\otimes b$ to $a\cdot b$ for all $a\in K[G]$ and $b\in I_H$ is an isomorphism of left K[G]-modules.

2. (2) The canonical map $K[G]\otimes_{K[H]} \left(I_H/I_T^H\right)\longrightarrow I_H^G/I_T^G$ that sends $a\otimes (b+I_T^H)$ to $ab+I_T^G$ for all $a\in K[G]$ and $b\in I_H$ is an isomorphism of left K[G]-modules.

3. (3) The kernel of the canonical map of K[G]-modules $K[G/T]\longrightarrow K[G/H]$ is naturally isomorphic to $I_H^G/I_T^G$ .

2.3 Hughes-free division rings and $L^2$ -Betti numbers

Let G be a locally indicable group and let K be a field. An embedding $\varphi \colon K[G] \hookrightarrow \mathcal D$ of the group algebra K[G] into a division ring $\mathcal D$ is called Hughes-free if the following conditions hold.

1. (1) The image $\varphi(K[G])$ generates $\mathcal D$ as a division ring.

2. (2) Let $H \leqslant G$ be a finitely generated subgroup and let $f \colon H \longrightarrow \mathbb{Z}$ be an epimorphism with kernel N, and let $t \in H$ map to a generator of $\mathbb{Z}$ under f. Let $\mathcal D_N$ denote the division closure of $\varphi(K[N])$ . Then $\{ \varphi(t^i) : i \in \mathbb{Z}\} \subseteq \mathcal D$ is linearly independent over $\mathcal D_N$ .

By a theorem of Hughes, if a Hughes-free embedding of K[G] exists, then it is unique up to K[G]-isomorphism [Reference HughesHug70]. Thus, if K[G] has a Hughes-free embedding, then we denote the division ring by $\mathcal{D}_{K[G]}$ and think of K[G] as a subset of $\mathcal{D}_{K[G]}$ . We will call $\mathcal{D}_{K[G]}$ the Hughes-free division ring of K[G]. Note that if $H \leqslant G$ is any subgroup, then the division closure of K[H] in $\mathcal{D}_{K[G]}$ is isomorphic to the Hughes-free division ring $\mathcal{D}_{K[H]}$ . The existence of Hughes-free division rings has been established for many classes of locally indicable groups, and, in particular, for all locally indicable groups when the ground field K has characteristic zero.

1. (1) the field K is of characteristic zero, or

2. (2) G is residually (locally indicable and amenable) or virtually compact special.

The groups we will be working with in this paper are locally indicable and virtually compact special, so we will always assume that any group algebra K[G] has a Hughes-free division ring $\mathcal{D}_{K[G]}$ .

Hughes-free division rings provide powerful homological invariants. Recall that modules over a division ring are automatically free modules and that they have a well-defined dimension. Thus, if M is a K[G]-module, we can define its $\mathcal{D}_{K[G]}$ -dimension by

\[ \dim_{\mathcal{D}_{K[G]}} M := \dim_{\mathcal{D}_{K[G]}} (\mathcal{D}_{K[G]} \otimes_{K[G]} M)\]

and more generally $\mathcal{D}_{K[G]}$ -Betti numbers by

(2.1) \begin{equation} \beta_n^{K[G]}(M) := \dim_{\mathcal{D}_{K[G]}} {\rm Tor}_n^{K[G]}(\mathcal{D}_{K[G]}, M).\end{equation}

We will not need these $\mathcal{D}_{K[G]}$ -Betti numbers of general K[G]-modules until § 6. Note that $\beta_0^{k[G]}(M) = \dim_{\mathcal{D}_{K[G]}} M$ . When $K = \mathbb{C}$ , we will write $\beta_n^{(2)}(M)$ instead of $\beta_n^{K[G]}(M)$ . Setting K to be the trivial K[G]-module, we obtain homological numerical invariants of the group G:

(2.2) \begin{equation} b_n^{K[G]}(G) := \beta_n^{K[G]}(K) = \dim_{\mathcal{D}_{K[G]}} {\rm Tor}_n^{K[G]}(\mathcal{D}_{K[G]}, K).\end{equation}

We will refer to these as the $\mathcal{D}_{K[G]}$ -Betti numbers of G.

The properties listed in the following proposition will be used throughout the paper. We emphasise point (1) below, which states that when $K = \mathbb{C}$ , the $\mathcal{D}_{K[G]}$ -Betti numbers coincide with the $L^2$ -Betti numbers of G.

1. (1) If $K= \mathbb{C}$ , then $b_n^{K[G]}(G) = b^{(2)}_n(G)$ for all integers $n\geq 0$ .

2. (2) If G is non-trivial, then $b_0^{K[G]}(G) = 0$ , otherwise $b_0^{K[G]}(G) = 1$ .

3. (3) If G is of finite type, then $\chi(G) = \sum_{i=0}^\infty (-1)^i b_i^{K[G]}(G)$ .

4. (4) Let $H \leqslant G$ be a subgroup of finite index. Then $\mathcal{D}_{K[H]}\otimes_{K[H]} K[G] \cong \mathcal{D}_{K[G]}$ as $(\mathcal{D}_{K[H]}, K[G])$ -bimodules. Consequently, $b_n^{K[H]}(H) = |G:H| \cdot b_n^{K[G]}(G)$ for all n.

5. (5) Let $k\geq 0$ be an integer, let H be a subgroup of G and let N be a left K[H]-module. There is an isomorphism of left $\mathcal{D}_{K[G]}$ -modules of the form

   \[{\rm Tor}_k^{K[H]}\big(\mathcal{D}_{K[G]}, N\big)\cong {\rm Tor}_k^{K[G]}\big(\mathcal{D}_{K[G]}, K[G]\otimes_{K[H] }N\big).\]
   In particular, if $N=K$ is the trivial K[H]-module, then
   \[b_k^{K[H]}(H)=\dim_{\mathcal{D}_{K[G]}} {\rm Tor}_k^{K[G]} \big(\mathcal{D}_{K[G]}, K[G/H]\big).\]

6. (6) Let $k\geq 0$ be an integer, let H be a finite-index subgroup of G, and let M be a left K[G]-module. There is an isomorphism of left $\mathcal{D}_{K[H]}$ -modules of the form

   \[{\rm Tor}_k^{K[H]}\big(\mathcal{D}_{K[H]}, M\big)\cong {\rm Tor}_k^{K[G]}\big(\mathcal{D}_{K[G]}, M\big).\]

7. (7) If G is free on $m \geqslant 1$ generators, then $b_1^{K[G]}(G) = m-1$ and $b_n^{K[G]}(G) = 0$ for all $n \neq 1$ . If G is amenable, then $b_n^{K[G]}(G) = 0$ for all n.

Proof. Statement 2.15 follows from [Reference Jaikin-Zapirain and López-ÁlvarezJ-ZL-Á20, Theorem 1.1], while (2) can easily be proven directly from the definitions. To prove (3), let

\[ 0 \longrightarrow K[G]^{r_d} \longrightarrow K[G]^{r_{d-1}} \longrightarrow \cdots \longrightarrow K[G]^{r_0} \longrightarrow K \longrightarrow 0 \]

be a resolution of the trivial K[G]-module K by finitely generated free K[G]-modules. By definition, $\chi(G) = \sum_{i=0}^d (-1)^i r_i$ . After tensoring with $\mathcal{D}_{K[G]}$ and omitting the rightmost term, we obtain the chain complex

\[ 0 \longrightarrow \mathcal{D}_{K[G]}^{r_d} \longrightarrow \mathcal{D}_{K[G]}^{r_{d-1}} \longrightarrow \cdots \longrightarrow \mathcal{D}_{K[G]}^{r_0} \longrightarrow 0, \]

whose boundary maps we denote by $\partial_i \colon \mathcal{D}_{K[G]}^{r_i} \longrightarrow \mathcal{D}_{K[G]}^{r_{i-1}}$ . Since $\mathcal{D}_{K[G]}$ is a division ring, the rank-nullity theorem holds, and therefore there is a decomposition

\[ \mathcal{D}_{K[G]}^{r_i} \cong \ker \partial_i \oplus \textrm{im} \partial_i \cong {\rm Tor}_i^{K[G]}(\mathcal{D}_{K[G]},K) \oplus \textrm{im} \partial_{i+1} \oplus \textrm{im} \partial_i. \]

Since, by definition, $b_i^{K[G]}(G) = \dim_{\mathcal{D}_{K[G]}}{\rm Tor}_i^{K[G]}(\mathcal{D}_{K[G]},K)$ , we obtain

\[\chi(G) = \sum_{i=0}^\infty (-1)^i b_i^{K[G]}(G).\]

Statement (4) is a direct consequence of [Grä20, Corollary 8.3] (for a detailed proof see [Reference FisherFis24, Lemma 6.3]). The isomorphism of (5) follows from a standard application of Shapiro’s lemma on the second entry of the Tor functor. The second equation of (6) follows from setting the trivial K[H]-module $N=K$ and from noting that, as left K[G]-modules, $K[G]\otimes_{K[H]}K\cong K[G/H]$ . Similarly, we apply Shapiro’s lemma to the first entry of Tor to obtain the isomorphism

\[ {\rm Tor}_k^{K[H]}(\mathcal{D}_{K[H]}, M)\cong {\rm Tor}_k^{K[G]}(\mathcal{D}_{K[H]}\otimes_{K[H]} K[G], M). \]

Now (6) follows from (4) and the previous isomorphism. Finally, for (7), the claim about free groups can be proved easily using (2) and (3). The claim about amenable groups follows from [Reference Henneke and KielakHK21, Theorem 3.9(6)] (only the case $K = \mathbb{Q}$ is treated there, but the case with K arbitrary has the same proof).

3.1 Definitions and basic properties

The notion of $L^2$ -independence and the $L^2$ -Hall property were introduced by Antolín and Jaikin-Zapirain [Reference Antolín and Jaikin-ZapirainAJ-Z22] in connection with proving that surface groups satisfy the SHNC. We recall these definitions.

Definition 3.2. Let H be a subgroup of G. Consider the natural surjection of left K[G]-modules $K[G/H]\longrightarrow K$ . This induces a natural map

\[ {\rm Tor}_1^{K[G]}(\mathcal{D}_{K[G]}, K[G/H]) \longrightarrow {\rm Tor}_1^{K[G]}(\mathcal{D}_{K[G]}, K). \]

We say that H is $\mathcal{D}_{K[G]}$ -independent if the map is injective. When $K = \mathbb{C}$ , we will say that H is $L^2$ -independent in G.

The injectivity of the above map depends on the choice of embedding of H into G. For example, the embedding $f\colon F(a, b, c)\longrightarrow G= F(x, y, z)$ defined by $f(a)=x^2$ , $f(b)=y$ , and $f(c)=y^x$ does not lead to an $L^2$ -independent subgroup of G. For this reason, the following definition will be useful later.

By [Reference Antolín and Jaikin-ZapirainAJ-Z22, Proposition 4.2], H is $\mathcal{D}_{K[G]}$ -independent in G if and only if the corestriction map $H_1(H;\mathcal{D}_{K[G]}) \longrightarrow H_1(G;\mathcal{D}_{K[G]})$ is injective. So Definition 3.2 is the natural generalisation of Antolín and Jaikin-Zapirain’s definition of $L^2$ -independence [Reference Antolín and Jaikin-ZapirainAJ-Z22, Section 4] for other division rings $\mathcal{D}_{K[G]}$ . Working in this greater generality will uniformly include various cases of interest while adding no technical difficulty.

The augmentation ideal corresponding to a subgroup captures a lot of structure of the subgroup and, hence, the following proposition provides a useful reformulation of the notion of $\mathcal{D}_{K[G]}$ -independence.

Remark 3.6. Note that the $L^2$ -Hall property can be defined for all groups, while the $\mathcal{D}_{K[G]}$ -Hall property only makes sense for locally indicable groups for which a Hughes-free embedding $K[G] \hookrightarrow \mathcal{D}_{K[G]}$ exists. Indeed, if $H \leqslant G$ and $\mathcal U(G)$ is the algebra of affiliated operators of G, then we say that H is $L^2$ -independent in G if

(3.1) \begin{equation} \dim_{\mathcal U(G)} \ker\left(H_1(H; \mathcal U(G)) \longrightarrow H_1(G; \mathcal U(G))\right) = 0, \end{equation}

and that G has the $L^2$ -Hall property if every finitely generated subgroup of G is $L^2$ -independent in a finite-index subgroup of G. These definitions agree with Definitions 3.2 and 3.5 by [Reference Antolín and Jaikin-ZapirainAJ-Z22, Lemma 4.1]. We mention this because we will discuss the $L^2$ -Hall property for some non-locally indicable groups later in this section. On the other hand, an advantage of working with the $\mathcal{D}_{K[G]}$ -Hall property is that the condition that $H_1(H;\mathcal{D}_{K[G]}) \longrightarrow H_1(G;\mathcal{D}_{K[G]})$ be injective is somewhat less awkward than the condition in (3.1).

The following hereditary feature of the $L^2$ -Hall property will be useful later. Recall that a ring homomorphism $R \longrightarrow S$ is (right) faithfully flat if for every morphism $M \longrightarrow N$ of (right) R-modules, $M \longrightarrow N$ is injective if and only if $M \otimes_R S \longrightarrow N \otimes_R S$ is injective. There is the corresponding concept of left faithful flatness, which is defined analogously. If $\mathcal D_1 \longrightarrow \mathcal D_2$ is a morphism of division rings, then it is necessarily injective and (left and right) faithfully flat. Indeed, consider a morphism of $\mathcal D_1$ -modules $M \longrightarrow N$ . Since $\mathcal D_1$ is a division ring, $\mathcal D_2 \cong \oplus_I \mathcal D_1$ for some index set I. From the commutative diagram

it follows at once that $M \longrightarrow N$ is injective if and only if $M \otimes_{\mathcal D_1} \mathcal D_2 \longrightarrow N \otimes_{\mathcal D_1} \mathcal D_2$ is.

is injective. But then $H_1(U;\mathcal{D}_{K[G_0]}) \longrightarrow H_1(G_0 \cap H; \mathcal{D}_{K[G_0]})$ is injective. Since extensions of division rings are faithfully flat, the commutative diagram

implies that $H_1(U; \mathcal{D}_{k[G_0 \cap H]}) \longrightarrow H_1(G_0 \cap H; \mathcal{D}_{k[G_0 \cap H]})$ is injective. Hence, H has the $L^2$ -Hall property.

We now collect various instances where we understand $L^2$ -independent subgraphs of groups. The first such instance is the following lemma, which will be useful when establishing the $L^2$ -Hall property for graphs of free groups with cyclic edge groups.

1. (1) the maps $Y_v\longrightarrow Z_v$ are $\mathcal{D}_{K[Z]}$ -injective for all $v\in \mathsf{Vert}(\Gamma^\mathcal Y)$ ,

2. (2) $b_1^{\mathcal{D}_{K[Z]}}(Z_e) = 0$ for all $e\in \mathsf{Edge}(\Gamma^\mathcal Z)$ , and

3. (3) the groups $Y_e$ and $Z_e$ are isomorphic for all $e\in \mathsf{Edge}(\Gamma^\mathcal Y)$

4. (4) then the canonical injection $\pi_1(\mathcal Y) \longrightarrow \pi_1(\mathcal Z)$ is $\mathcal{D}_{K[Z]}$ -injective.

Since Chiswell’s Mayer–Vietoris exact sequence is induced by applying a Tor functor to the short exact sequences of the above form (see the proof of [Reference ChiswellChi76, Theorem 2]), the long exact sequences are automatically natural and thus we obtain maps between Chiswell’s exact sequences for $\mathcal Y$ and $\mathcal Z$

where $H_i(-)$ stands for $H_i(-;\mathcal D_{K[Z]})$ (for $i = 0,1$ ). By the four lemma, the map $H_1(\pi_1(\mathcal Y)) \longrightarrow H_1(Z)$ is injective.

The following technical proposition will be crucial to establishing the $L^2$ -Hall property for limit groups in § 5. We also consider it to be of potential interest for proving that relatively hyperbolic groups with a finite abelian hierarchy have the $L^2$ -Hall property (Conjecture 1.3).

Proposition 3.9. Let $\mathcal W$ be a subgraph of groups of $\mathcal Z$ that have the same underlying graph $\Gamma$ and all of whose edge groups are infinite cyclic. Let $G = \pi_1(\mathcal Z)$ and suppose that there is a bipartite structure $\mathsf{Vert}(\Gamma) = \mathsf{Vert}_\mathsf{o} \sqcup \mathsf{Vert}_\mathsf{t}$ of $\Gamma$ so that no two different edges of $\mathsf{Edge}(\Gamma)$ have the same endpoints. We assume, moreover, that the orientation on $\Gamma$ is such that $\mathsf{o}(e)\in \mathsf{Vert}_{\mathsf{o}}$ and $\mathsf{t}(e)\in \mathsf{Vert}_{\mathsf{t}}$ for all $e\in \mathsf{Edge}(\Gamma)$ . We denote by $z_{e}$ a generator of the infinite cyclic group $Z_e$ . Let T be a spanning tree of $\Gamma$ . Fix a presentation of G (as described in Definition 2.3) and, for every $e\in \mathsf{Edge}(\Gamma)\smallsetminus \mathsf{Edge}(T)$ , denote by $t_e$ the formal letter associated to e. For all $v\in \mathsf{Vert}_{\mathsf{o}}$ , we consider finite subsets $\mathcal L_v^{(0)}\subseteq\mathcal L_v\subseteq Z_v$ such that

\[ \mathcal{L}_v \setminus \mathcal{L}_v^{(0)}\subseteq \bigcup_{\mathsf{o}(e)=v} \phi_{\mathsf{o}, e}(z_e) \subseteq W_v\cup \big(\mathcal{L}_v \setminus \mathcal{L}_v^{(0)}\big). \]

Suppose that, for all $v\in \mathsf{Vert}_{\mathsf{o}}$ , the natural map

\[ W_v*\bigg(\coprod_{\mathcal L_v} \mathbb{Z} \bigg)\longrightarrow Z_v \]

is injective and $\mathcal{D}_{K[G]}$ -injective. We fix a subset $ \mathsf{Edge}(T)\subseteq E_T\subseteq \mathsf{Edge}(\Gamma)$ such that $\phi_{e, \mathsf{o}}(z_e)\in \mathcal L_{\mathsf{o}(e)}\setminus \mathcal L^{(0)}_{\mathsf{o}(e)}$ for all $e\in \mathsf{Edge}(\Gamma)\setminus E_T$ . If we name $ \mathcal L^{(0)}=\bigcup_{v\in \mathsf{Vert}(\Gamma)} \mathcal L_v^{(0)}$ and $\mathcal L^{(t)}=\{t_e: e\in \mathsf{Edge}(\Gamma)\setminus E_T\}$ , then the natural map

(3.2) \begin{equation} \pi_1(\mathcal W)* \bigg(\coprod_{\mathcal L^{(0)}\bigcup \mathcal L^{(t)}} \mathbb{Z}\bigg)\longrightarrow \pi_1 (\mathcal Z) \end{equation}

is injective and $\mathcal{D}_{K[G]}$ -injective (in the sense of Definition 3.3).

Proof. Recall from Lemma 2.5 that, given a subgraph of groups $\mathcal H$ of $\mathcal G$ , the canonical map $\pi_1(\mathcal H)\longrightarrow \pi_1(\mathcal G)$ injective. We will consider several intermediate graphs of groups $\mathcal W \leqslant \mathcal W^{(1)} \leqslant \mathcal W^{(2)} \leqslant \mathcal W^{(3)} \leqslant \mathcal W^{(4)} \leqslant \mathcal Z$ to prove the claim. We will only specify their vertex and edge groups, and the corresponding edge maps will be assumed to be the restrictions of the edge maps of $\mathcal Z$ .

The graph of groups $\mathcal W^{(1)}$ is defined as follows. For all $v\in \mathsf{Vert}_{\mathsf{t}}$ , $W_v^{(1)}=Z_v$ . For all $v\in \mathsf{Vert}_{\mathsf{t}}$ and $e\in \mathsf{Edge}(\Gamma)$ , $W_v^{(1)}=W_v$ and $W_e^{(1)}=W_e$ . By Lemma 3.8, the canonical map

(3.3) \begin{equation} \pi_1(\mathcal W)\longrightarrow \pi_1(\mathcal W^{(1)})\end{equation}

is $\mathcal{D}_{K[G]}$ -injective.

We split $\mathcal L_v\setminus \mathcal L_v^{(0)}$ as a disjoint union of $\mathcal L_v^{(1)}$ and $\mathcal L_v^{(2)}$ , where $\mathcal L_v^{(1)}$ consists exactly of the elements $\phi_{\mathsf{o}, e}(z_e)\in \mathcal L_v$ such that $e\in E_T$ . Consider another intermediate graph of groups $\mathcal W^{(1)} \leqslant \mathcal W^{(2)} \leqslant \mathcal Z$ defined as follows:

• − $W_v^{(2)}=W_v^{(1)}*\big(\coprod_{\mathcal{L}_v^{(1)}} \mathbb{Z}\big)$ for $v\in \mathsf{Vert}_{\mathsf{o}}$ ;

• − $W_v^{(2)}=W_v^{(1)}$ for $v\in \mathsf{Vert}_{\mathsf{t}}$ ;

• − $W_e^{(2)}=Z_e$ for $e\in E_T$ ;

• − $W_e^{(2)}=W_e^{(1)}$ for $e\in \mathsf{Edge}(\Gamma)\smallsetminus E_T$ .

Letting $E^{(1)}=\{t_e : e\in E_T\smallsetminus E(T), \phi_{e, \mathsf{o}}(z_e)\in \mathcal L_v^{(1)}\}$ , the canonical map

(3.4) \begin{equation} \pi_1(\mathcal W^{(1)})*\bigg( \coprod_{E^{(1)}} \mathbb{Z} \bigg) \longrightarrow \pi_1(\mathcal W^{(2)}) \end{equation}

is an isomorphism, so $\pi_1(\mathcal W^{(1)})\longrightarrow \pi_1(\mathcal W^{(2)})$ is $\mathcal{D}_{K[G]}$ -injective.

We define $\mathcal W^{(2)} \leqslant \mathcal W^{(3)}\leqslant \mathcal Z$ as follows:

• − $W_v^{(3)}=W_v^{(2)}*\big(\coprod_{\mathcal{L}_v^{(0)}} \mathbb{Z}\big)$ for $v\in \mathsf{Vert}_{\mathsf{o}}$ ;

• − $W_v^{(3)}=W_v^{(2)}$ for $v\in \mathsf{Vert}_{\mathsf{t}}$ ;

• − $W_e^{(3)}=W_v^{(2)}$ for $e\in E_T$ ;

• − $W_e^{(3)}= W_v^{(2)}$ for $e\in \mathsf{Edge}(\Gamma)\smallsetminus E_T$ .

It is immediate from the presentation of $\pi_1(\mathcal W^{(3)})$ that the canonical map

(3.5) \begin{equation} \pi_1(\mathcal W^{(2)})*\bigg(\coprod_{\mathcal L^{(0)}} \mathbb{Z}\bigg)\longrightarrow \pi_1(\mathcal W^{(3)}) \end{equation}

is an isomorphism. Finally, we define $\mathcal W^{(3)} \leqslant \mathcal W^{(4)} \leqslant \mathcal Z$ as follows:

• − $W_v^{(4)}=W_v^{(3)}*\big(\coprod_{\mathcal{L}_v^{(2)}} \mathbb{Z}\big)$ for $v\in \mathsf{Vert}_{\mathsf{o}}$ ;

• − $W_v^{(4)}=W_v^{(3)}$ for $v\in \mathsf{Vert}_{\mathsf{t}}$ ;

• − $W_e^{(4)}=Z_e$ for $e\in \mathsf{Edge}(\Gamma)$ .

  We observe that

  (3.6) \begin{equation} \pi_1(\mathcal W^{(3)})*\bigg(\coprod_{\mathcal L^{(t)}} \mathbb{Z}\bigg)\longrightarrow \pi_1(\mathcal W^{(4)}) \end{equation}
  is an isomorphism. Observe that $ \mathcal W^{(4)} \leqslant \mathcal Z$ admits the following description:

• − $W_v^{(4)}=W_v *\left(\coprod_{\mathcal{L}_v} \mathbb{Z}\right)$ for $v\in \mathsf{Vert}_{\mathsf{o}}$ ;

• − $W_v^{(4)}=Z_v$ for $v\in \mathsf{Vert}_{\mathsf{t}}$ ;

• − $W_e^{(4)}=Z_e$ for $e\in \mathsf{Edge}(\Gamma)$ .

By our assumption on (3.2) and by Lemma 3.8, the canonical map

(3.7) \begin{equation} \pi_1(\mathcal W^{(4)})\longrightarrow \pi_1(\mathcal Z) \end{equation}

is $\mathcal{D}_{K[G]}$ -injective. From the chain of injections and $\mathcal{D}_{K[G]}$ -injections described in (3.3)– (3.7) we conclude that the canonical map

\[ \pi_1(\mathcal W)* \bigg(\coprod_{\mathcal L^{(0)} \bigcup \mathcal L^{(t)}} \mathbb{Z}\bigg)\longrightarrow \pi_1 (\mathcal Z) \]

is injective and $\mathcal{D}_{K[G]}$ -injective. The proof is complete.

3.2 Examples

We are already in a position to establish the $\mathcal{D}_{K[G]}$ -Hall property for some classes of groups.

There are also non-amenable groups which are $L^2$ -Hall for the reason discussed above. As an example, let T be a Tarski monster of prime order p and let $G = T \times \mathbb{Z}$ . Since all the proper subgroups of T are isomorphic to $\mathbb{Z}/p$ , it follows that every finitely generated subgroup H of G has $b^{(2)}_1(H) = 0$ and therefore G is $L^2$ -Hall. However, T is non-amenable, and therefore so is G. Note that G is not locally indicable (or even torsion-free) and therefore $\mathcal{D}_{K[G]}$ does not exist. However, it still makes sense to discuss the $L^2$ -Hall property for this group since $L^2$ -invariants are defined for all groups.

Fundamental groups of closed surfaces also satisfy a principle analogous to Hall’s theorem, namely that finitely generated subgroups are virtual retracts, as proved by Scott [Reference ScottSco78] using hyperbolic geometry (see also [Reference WiltonWil07] for a more combinatorial proof). This directly implies that surface groups are subgroup separable. Moreover, Antolín and Jaikin-Zapirain proved that they are $L^2$ -Hall in [Reference Antolín and Jaikin-ZapirainAJ-Z22, Theorem 4.4] using these virtual retractions combined with other algebraic ideas (such as the theory of Demushkin groups and the cohomological goodness of surface groups). We now use Scott’s argument to give a more topological proof of the $L^2$ -Hall property for surface groups.

Proof of Proposition 3.12. We say that a compact connected subsurface X of a connected surface S is incompressible if no component of the closure of the complement $S\smallsetminus X$ is a disc. If $\pi_1(X) \neq 1$ , then X is incompressible if and only if the induced map $\pi_1(X)\longrightarrow \pi_1(S)$ is injective.

Let G be the fundamental group of a closed connected surface $\Sigma$ with $\chi(\Sigma)\leqslant 0$ (the case when $\chi(\Sigma)>0$ is trivial). Let $H \leqslant G$ be a non-trivial finitely generated subgroup. Let $\Sigma' \longrightarrow \Sigma$ be the covering space corresponding to H. Then $\Sigma'$ is a (possibly non-compact) surface with fundamental group H. Let $\Sigma_c $ be a compact core for $\Sigma'$ , that is, $\Sigma_c \subseteq \Sigma'$ is a compact, connected, incompressible subsurface such that the natural map $\pi_1(\Sigma_c)\longrightarrow \pi_1(\Sigma')$ is an isomorphism. The existence of $\Sigma_c$ is ensured by [Reference ScottSco78, Lemma 1.5]. Scott also showed in [Reference ScottSco78, Lemma 1.4 and Theorem 3.3] that there is a commutative diagram

where $\widehat \Sigma \longrightarrow \Sigma$ is an intermediate finite-sheeted covering into which $\Sigma_c$ projects homeomorphically. Since $ \pi_1 (\Sigma_c)\cong H\neq 1$ , the boundary $\partial \Sigma_c$ is incompressible in $ \Sigma_c$ . Consequently, $\Sigma_c$ is an incompressible subsurface of $\widehat \Sigma$ and every connected component $\widehat \Sigma_i$ of the closure of the complement $\widehat\Sigma \smallsetminus \Sigma_c$ has the property that its boundary is incompressible. It follows that $\widehat \Sigma$ admits a decomposition as a finite graph of spaces where the vertex spaces are $\{\widehat \Sigma_i, \Sigma_c\}$ , various of which are glued along some of their boundary components (so the edge spaces are circles and the edge maps are $\pi_1$ -injective). This produces a splitting for the fundamental group $\pi_1 (\widehat \Sigma)$ where one vertex is $\pi_1(\Sigma_c)$ , the other vertices are $\pi_1(\widehat \Sigma_i)$ and the edge groups are infinite cyclic. By Lemma 3.8, the group $H=\pi_1(\Sigma_c)$ is $\mathcal{D}_{K[G]}$ -independent in $\pi_1\big(\widehat \Sigma\big)$ , and therefore G is $\mathcal{D}_{K[G]}$ -Hall.

The ideas of Proposition 3.12, such as the construction of a compact core for a subgroup H and the reconstruction of H from cyclic splittings, motivates the strategy that we follow in Theorem 4.9 for more general graphs of free groups with cyclic edges. We can now explain the simpler case when the edge groups are trivial (i.e. the case of free products), which generalises the proof that free groups are $\mathcal{D}_{K[G]}$ -Hall.

Let $H \leqslant A * B$ be a finitely generated subgroup and let $Y \longrightarrow X$ be the covering space corresponding to H. Let Z be a finite core for Y, that is, $Z \hookrightarrow Y$ induces a $\pi_1$ -isomorphism, the underlying graph of Z is finite, and $Z_v = Y_v$ for all vertices v in the underlying graph of Z. Denote the fundamental groups of the A-vertices (i.e. those vertex spaces in Z covering $X_A$ ) by $X_{A_i}$ , where $\pi_1(X_{A_i}) = A_i \leqslant A$ . Similarly, denote the B-vertices by $X_{B_j}$ , where $\pi_1(X_{B_j}) = B_j \leqslant B$ . For each i (respectively, j), let $A_i' \leqslant A$ (respectively, $B_j' \leqslant B$ ) be a finite-index subgroup containing $A_i$ (respectively, $B_j$ ) such that $A_i \hookrightarrow A_i'$ (respectively, $B_j \hookrightarrow B_j'$ ) is $\mathcal{D}_{K[G]}$ -injective (recall Definition 3.3).

Let $X_{A_i} \longrightarrow X_{A_i'}$ be the covering map associated to $A_i \leqslant A_i'$ and let $P_i \subseteq X_{A_i}$ be the set of points that are the endpoints of edges Z. By subgroup separability of A, we may find a finite-index subgroup $A_i''$ such that $A_i \leqslant A_i'' \leqslant A_i'$ and such that the induced covering map $X_{A_i} \longrightarrow X_{A_i''}$ is injective on $P_i$ . Note that $A_i$ is still $\mathcal{D}_{K[G]}$ -injective in $A_i''$ . A similar discussion applies to the B-vertices, where we obtain new groups $B_j''$ and spaces $X_{B_j''}$ satisfying the analogous conditions.

Let $\overline Z$ be the following graph of spaces: it has the same underlying as Z, the vertex spaces $X_{A_i}$ (respectively, $B_j$ ) are replaced with $X_{A_i''}$ (respectively, $X_{B_j''}$ ), and there is an edge joining the points $x \in X_{A_i''}$ and $y \in X_{B_j''}$ if and only if they are the images of points x’ and y’ under the coverings $X_{A_i} \longrightarrow X_{A_i''}$ and $X_{B_j} \longrightarrow X_{B_j''}$ respectively, and x’ and y’ are joined by an edge in Z. From the construction, the covering spaces of the vertices induce a map of graphs of spaces $Z \longrightarrow \overline Z$ (which is an isomorphism on underlying graphs). Then $\pi_1(Z) \hookrightarrow \pi_1(\overline Z)$ is $\mathcal{D}_{K[G]}$ -injective by Lemma 3.8.

The process of completing $\overline Z$ to a finite-sheeted cover $\widehat Z$ of X is standard. This is detailed, for instance, in [Reference WiltonWil07, Theorem 3.2]. For this, one adds various disjoint copies of the vertices $X_A$ and $X_B$ to the precover $\overline Z$ until the resulting space satisfies Stallings’ principle (see [Reference WiltonWil07, Proposition 3.1]). Then certain pairs of the hanging elevations of edge maps can be glued together along additional trivial edge spaces to produce the finite-sheeted cover $\overline Z \longrightarrow X$ . As before, the inclusion $\overline Z \hookrightarrow \widehat Z$ induces a $\mathcal{D}_{K[G]}$ -injection on fundamental groups, which proves the claim.

It is natural to ask whether subgroup separability is needed in Proposition 3.14, but it is unclear to the authors if, for instance, the free product of finitely generated and residually finite $L^2$ -Hall groups is $L^2$ -Hall. For non-residually finite groups we make the following observation.

We conclude with some non-examples.

3.3 Passing to finite-index overgroups

In this subsection, we prove Theorem 3.17. This will be crucial when establishing the $L^2$ -Hall property for graphs of free groups with cyclic edge groups. Theorem D from the introduction will follow from Corollary 3.20 and Lemma 3.21.

1. (1) If H is $\mathcal{D}_{K[G]}$ -independent in G, then $H\cap G_1$ is $\mathcal{D}_{K[G]}$ -independent in $G_1.$

2. (2) If there exists a finite-index subgroup $H_0\leqslant H$ such that $H_0$ is $\mathcal{D}_{K[G]}$ -independent in $G_1$ , then there exists a finite-index subgroup $G_0\leqslant G$ containing H as a $\mathcal{D}_{K[G]}$ -independent subgroup.

Statement (1) is essentially [Reference Antolín and Jaikin-ZapirainAJ-Z22, Proposition 5.2], whose argument is followed to additionally prove statement (2). We first prove the following simple lemma.

Proof. Consider the short exact sequence of K[G]-modules

\[ 0\longrightarrow I_{T}^G/I_H^G\longrightarrow I_G/I_H^G\longrightarrow I_G/I_T^G\longrightarrow 0. \]

The induced long exact sequence in ${\rm Tor}_\bullet^{K[G]}(\mathcal{D}_{K[G]}, -)$ contains the following sequence of $\mathcal{D}_{K[G]}$ -modules:

\[ {\rm Tor}_1^{K[G]}\big(\mathcal{D}_{K[G]}, I_G/I_H^G\big)\longrightarrow {\rm Tor}_1^{K[G]}\big(\mathcal{D}_{K[G]}, I_G/I_T^G\big) \longrightarrow {\rm Tor}_0^{K[G]}\big(\mathcal{D}_{K[G]}, I_{T}^G/I_H^G\big). \]

By Proposition 3.4 and the assumption that H is $\mathcal{D}_{K[G]}$ -independent in G, it follows that the leftmost term ${\rm Tor}_1^{K[G]}\big(\mathcal{D}_{K[G]}, I_G/I_H^G\big)$ is zero. Moreover, since H is finite index in T, it is not hard to see that $I_{T}^G/I_H^G$ is a finite-dimensional K-vector space, so ${\rm Tor}_0^{K[G]}\big(\mathcal{D}_{K[G]}, I_{T}^G/I_H^G\big)=0$ . It follows directly from the short exact sequence above that ${\rm Tor}_1^{K[G]}\big(\mathcal{D}_{K[G]}, I_G/I_T^G\big)=0$ . This implies, again by Proposition 3.4, that T is $\mathcal{D}_{K[G]}$ -independent in G.

We are now ready to explain the proof of Theorem 3.17.

Proof of Theorem 3.17. Let $H_1=H\cap G_1$ . We begin by proving statement (1). By Proposition 3.4, it is enough to show that

\[ {\rm Tor}_1^{K[G_1]}\big(\mathcal{D}_{K[G_1]}, I_{G_1}/I_{H_1}^{G_1}\big)=0.\]

The horizontal arrows $\iota_1$ and $\iota_2$ are injective. It is clear that $I_{H_1}^{G_1}\subseteq I_{G_1}\cap I_{H}^{G}.$ For the reverse inclusion, we will use the above diagram. If $x\in I_{G_1}\cap I_{H}^{G} $ , then $x\in K[G_1]$ and x belongs to the kernel of $p_H^G\circ \iota_1$ . By the commutativity of the diagram and the injectivity of $\iota_2$ , the element x must belong to the kernel of $p_{H_1}^{G_1}$ , which equals $I_{H_1}^{G_1}$ by Lemma 2.11.

Claim 3.19 implies that the natural map of $K[G_1]$ -modules $I_{G_1}/I_{H_1}^{G_1}\longrightarrow I_{G}/I_{H}^{G}$ is injective. Furthermore, since $I_{G_1}/I_{H_1}^{G_1}$ (respectively, $I_{G}/I_{H}^{G}$ ) is the kernel of the augmentation $K[G_1/H_1]\longrightarrow K$ (respectively, $K[G/H]\longrightarrow K$ ) by Lemma 2.11, there is an exact sequence of $K[G_1]$ -modules of the form

\[ 0\longrightarrow I_{G_1}/I_{H_1}^{G_1} \longrightarrow I_{G}/I_{H}^{G} \longrightarrow K[G/H]/K[G_1/H_1] \longrightarrow 0.\]

Let $T\subseteq G$ be a set of representatives for the double $(G_1, H )$ -cosets in G such that $1\in T$ . Denote by $M_t$ the $K[G_1]$ -module $K[G_1/(H^t\cap G_1)]$ . Then $K[G/H]\cong \oplus_{t\in T} M_t$ as $K[G_1]$ -modules. Let $\mathcal{D}=\mathcal{D}_{K[G_1]}$ . Notice that ${\rm Tor}_2^{K[G_1]}(\mathcal{D}, M_t)=0$ for all $t \in T$ . The reason is that, by Proposition 2.15(5), its $\mathcal{D}$ -dimension equals

(3.8) \begin{equation} b_2^{K[H^t\cap G_1]}(H^t\cap G_1)=b_2^{K[H]}(H) \cdot |H^t: H^t\cap G_1| = 0.\end{equation}

Note that, to obtain (3.8), we have also used the fact that $H^t\cap G_1$ is finite index in $H^t\cong H$ , as well as the multiplicativity of $\mathcal{D}_{K[G]}$ -Betti numbers (Proposition 2.15(4)). By the additivity of the $\mathcal{D}$ -dimension function, it follows from Proposition 2.15(5) and Equation 3.8 that the $K[G_1]$ -module $N=K[G/H]/K[G_1/H_1]\cong \oplus_{t\in T \smallsetminus \{1\}} M_t$ has ${\rm Tor}_2^{K[G_1]}(\mathcal{D}, N)=0$ .

The long exact sequence in Tor gives us an exact sequence of $\mathcal{D}$ -modules of the form

(3.9)

We have already proved that ${\rm Tor}_2^{K[G_1]}(\mathcal{D}, N)=0$ . So statement (1) will follow from diagram (3.9) if we prove that ${\rm Tor}_1^{K[G_1]}\big(\mathcal{D}, I_{G}/I_{H}^{G}\big)=0$ . We know from Proposition 2.15(6) that

\begin{equation*} {\rm Tor}_1^{K[G_1]}\big(\mathcal{D}, I_{G}/I_{H}^{G}\big) \cong {\rm Tor}_1^{K[G]}\big(\mathcal{D}_{K[G]}, I_{G}/I_{H}^{G}\big).\end{equation*}

Furthermore, the right-hand side vanishes by Proposition 3.4 and the assumption that H is $\mathcal{D}_{K[G]}$ -independent in G. This completes the proof of statement (1).

We now prove (2). The subgroups $H_0\leqslant H\cap G_1\leqslant G_1$ have the property that $| H\cap G_1: H_0|<\infty$ and that $H_0$ is $\mathcal{D}_{K[G]}$ -independent in $G_1$ . By Lemma 3.18, $H\cap G_1$ is $\mathcal{D}_{K[G]}$ -independent in $G_1$ . Let $G_2 \trianglelefteq G_1$ be a normal subgroup of finite index; since $b^{K[H]}_2(H)=0$ , it follows from part (1) that $H\cap G_2$ is $\mathcal{D}_{K[G]}$ -independent in $G_2$ . Thus, by replacing $G_1$ by $G_2$ , we may assume that $G_1$ is normal in G and that $H\cap G_1$ is $\mathcal{D}_{K[G]}$ -independent in $G_1$ .

We claim that H is $\mathcal{D}_{K[G]}$ -independent in $G_0=\langle G_1, H\rangle=G_1\cdot H$ . For this, we first observe that $T=\{1\}$ is a set of representatives of the double $(G_1, H)$ -cosets in $G_0$ . So the argument given in (1) shows that the canonical map

(3.9) \begin{equation} I_{G_1}/I_{H\cap G_1}^{G_1}\xrightarrow[]{\cong} I_{G_0}/I_{H}^{G_0}\end{equation}

is an isomorphism of $K[G_1]$ -modules. Using that $H\cap G_1$ is $\mathcal{D}_{K[G]}$ -independent in $G_1$ , we can argue as before to deduce from (3.9) and Propositions 2.15(6) and 3.4 that

$$ {\rm Tor}_1^{K[G_0]}\big(\mathcal{D}_{K[G_0]}, I_{G_0}/I_{H}^{G_0}\big)\cong{\rm Tor}_1^{K[G_1]}\big(\mathcal{D}_{K[G_1]}, I_{G_1}/I_{H\cap G_1}^{G_1}\big)=0.$$

Thus, again by Proposition 3.4, H is $\mathcal{D}_{K[G]}$ -independent in $G_0$

The following result is a direct consequence of Theorem 3.17.

Corollary 3.20 offers a more flexible reformulation of the $\mathcal{D}_{K[G]}$ -Hall property which will be used in Theorem 4.9 to establish the $\mathcal{D}_{K[G]}$ -Hall property for various graphs of free groups and cyclic edge groups. Moreover, the local condition on the vanishing of $b^{K[H]}_2(H)=0$ for all $H \leqslant G$ can be condensed for certain groups of cohomological dimension 2 using the following lemma.

Proof. Note that the natural map

(3.10) \begin{equation} \mathcal{D}_{K[H]} \otimes_{K[H]} F \longrightarrow \mathcal{D}_{K[G]} \otimes_{K[G]} F \end{equation}

is injective, where F is a free left K[G]-module. To see this, it is enough to prove the claim when $F =K[G]$ . Let T be a right transversal for H in G. The Hughes-freeness of $\mathcal{D}_{K[G]}$ implies that the map $\oplus_{t \in T} \mathcal{D}_{K[H]} \cdot t \longrightarrow \mathcal{D}_{K[G]}$ induced by the inclusions $\mathcal{D}_{K[H]} \cdot t \hookrightarrow \mathcal{D}_{K[G]}$ is injective [Grä20, Corollary 8.3]. The map of (3.11) when $F=K[G]$ equals the composition

\[ \mathcal{D}_{K[H]} \otimes_{K[H]} K[G] \xrightarrow{\cong} \mathcal{D}_{K[H]} \otimes_{K[H]} \bigg( \bigoplus_{t \in T} K[H] \cdot t \bigg) \xrightarrow{\cong}\bigoplus_{t \in T} \mathcal{D}_{K[H]} \cdot t \hookrightarrow \mathcal{D}_{K[G]} \]

and is therefore injective, as desired.

The claim now follows easily. Let $0 \longrightarrow F_n \longrightarrow \cdots \longrightarrow F_0 \longrightarrow K \longrightarrow 0$ be a free resolution of the trivial K[G]-module K. This resolution exists because G has a classifying space of dimension at most n (we do not claim the modules $F_i$ to be finitely generated). If $b^{K[H]}_n(H) \neq 0$ , then there is a non-trivial element z in the kernel of $\mathcal{D}_{K[H]} \otimes_{K[H]} F_n \longrightarrow \mathcal{D}_{K[H]} \otimes_{K[H]} F_{n-1}$ . Then z is also a non-zero element of the kernel of $\mathcal{D}_{K[G]} \otimes_{K[G]} F_n \longrightarrow \mathcal{D}_{K[G]} \otimes_{K[G]} F_{n-1}$ and therefore $b^{K[G]}_n(G) \neq 0$ .

While we only have a conjectural characterisation of which general graphs of free groups with cyclic edge are $L^2$ -Hall, the case of an amalgam is entirely understood.

4.1 Proof of Theorem A

We prove Theorem 4.9 above (which is Theorem A from the introduction). We make a few simplifying reductions that we hope will make the visualisation of the objects easier as well as put us in the context of the proof of [Reference WiseWis00, Theorem 5.2].

Note that an immersed Klein bottle K is indeed the image of a Klein bottle surface S under a cellular immersion, where S is the generalised Baumslag–Solitar complex associated to the loop corresponding to K. Similarly, if T is an immersed torus, then there exist a topological torus S and a cellular immersion $S \longrightarrow T$ .

Proof. By Claim 4.13, we may assume that X is clean. Let $K_1, \ldots, K_n$ denote the immersed Klein bottles in X. By subgroup separability of $G = \pi_1(X)$ , there is a finite-sheeted regular cover $p_1 \colon X_1 \longrightarrow X$ where the components of $p^{-1}(K_1)$ are all immersed tori. Assume now that we have constructed some finite-sheeted regular cover $p_i \colon X_i \longrightarrow X$ so that for each $j \leqslant i$ every component of $p_i^{-1}(K_j)$ is an immersed torus. Let K be an immersed Klein bottle component of $p_i^{-1}(K_{i+1})$ . Again we may pass to a further finite-sheeted cover $q_{i+1} \colon X_{i+1} \longrightarrow X_i$ such that the composition

\[ p_{i+1} \colon X_{i+1} \longrightarrow X_i \longrightarrow X \]

is regular and every component of $q_{i+1}^{-1}(K)$ is an immersed torus. But then every component of $p_{i+1}^{-1}(K_{i+1})$ is an immersed torus by regularity of the cover. By Corollary 3.20, it is enough to show that $\pi_1(X_n)$ has the $L^2$ -Hall property.

Proof (of Theorem 4.9). By the claims above, we assume without loss of generality that G is the fundamental group of a clean graph of spaces $X = (X_v, X_e; \Gamma)$ , where the vertex spaces $X_v$ are graphs and the edge spaces $X_e$ are circles. Moreover, we assume that X does not contain any immersed Klein bottles.

Let H be a finitely generated subgroup of G. Following the proof of [Reference WiseWis00, Theorem 5.2], we will show that there is a finite-index subgroup $H_1 \leqslant H$ that is $L^2$ -independent in a finite-index subgroup $G_1 \leqslant G$ . By Theorem 3.17, this is enough to prove the theorem. We break up our proof into steps in a similar way to the proof of [Reference WiseWis00, Theorem 5.2].

Let $Y \longrightarrow X$ be the covering space corresponding to H. Note that Y has a natural decomposition into a clean graph of spaces $(Y_v, Y_e; \Gamma_Y)$ , where each of the vertex spaces are graphs and each of the edge spaces are homeomorphic to either $S^1$ or $\mathbb{R}$ .

Step 1 (The subcomplex). Denote the underlying graph of Y by $\Gamma(Y)$ . Since H is finitely generated, there is a finite connected subgraph $\Upsilon$ of $\Gamma(Y)$ such that the inclusion $Y_\Upsilon \hookrightarrow Y$ of the restricted graph of spaces $Y_\Upsilon$ is a $\pi_1$ -isomorphism.

Step 2 (Pruning). For each vertex space $Y_v$ of $Y_\Upsilon$ , let $Z_v$ be the smallest connected subgraph containing the images of all the edge spaces of $Y_\Upsilon$ and such that $Z_v \hookrightarrow Y_v$ induces a $\pi_1$ -isomorphism. Let $Z \subseteq Y_\Upsilon$ be the union of the spaces $Z_v$ and the edge spaces $Y_e$ of $Y_\upsilon$ . Note that Z is connected and has a natural graph of spaces structure $(Z_v, Z_e = Y_e; \Upsilon)$ such that the inclusion $Z \hookrightarrow Y_\Upsilon$ still induces a $\pi_1$ -isomorphism. The resulting vertex spaces of $Z_v$ have a compact core with pairs of infinite rays attached to them coming from the attaching maps of non-compact edge spaces in Y (see Figure 1).

Figure 1. A vertex space of Z. The thickened lines represent attaching maps of non-compact edge spaces, each of which being homeomorphic to $\mathbb{R}$ .

Step 3 ( $L^2$ -independence of periphery closing). This is the main step of the proof. Let e be an edge of $\Upsilon$ and let $I = [0,1]$ be the closed unit interval. If $Z_e \cong S^1$ , then we call $Z_e \times I$ a cylinder; if $Z_e \cong \mathbb{R}$ , then $Z_e \times I$ is a strip. If two strips in Z have a non-compact intersection, then the periodicity of the attaching maps implies that their intersection must in fact be homeomorphic to $\mathbb{R}$ .

Note that $\mathbb{Z}$ acts on each of the strips by covering translations (where the covering refers to a strip in Z covering a cylinder in X). As in [Reference WiseWis00, Theorem 5.2, Step 3], choose n large enough so that, for any edge e corresponding to a strip, all the vertices of $Z_e \times I$ with valence at least 3 (the valence is counted in the vertex graphs adjacent to the strip) are a distance less than n apart. Then quotient the strips of Z by the action of $n\mathbb{Z}$ to form a new complex $A = Z / {\sim}$ . Now A is a clean compact graph of graphs with $S^1$ edge spaces. A typical vertex space is shown in Figure 2.

Figure 2. A vertex space of A. The vertex space is obtained from that in Figure 1 by quotienting the thickened lines by the action of $n\mathbb{Z}$ .

We need to introduce a definition based on one given by Hsu and Wise [Reference Hsu and WiseHW10, Definition 9.1]. Declare two strips to be equivalent if their intersection is a copy of $\mathbb{R}$ ; this rule generates an equivalence relation on the set of strips in Z. The periphery containing a strip S is the union of the strips in the equivalence class of S.

Fix a periphery P. By the assumption that X contains no immersed Klein bottles, there is a compact subset $K \subseteq Z$ such that $P \smallsetminus K$ is homeomorphic to two disjoint copies of $\mathbb{R}_{>0} \times \Omega$ , where $\Omega$ is some finite graph. The effect of quotienting by the action of $n\mathbb{Z}$ can then be rephrased as follows. Choose K compact and sufficiently large so that K is a compact core for Z and all the vertices in $P \smallsetminus K$ are of degree 2. We also require, for every strip $S \subseteq P$ , that $K \cap S$ be a fundamental domain for the action of $n\mathbb{Z}$ on S. Denote by $R_1$ and $R_2$ the copies of $\mathbb{R}_{>0} \times \Omega$ in $P \smallsetminus K$ . We then form the quotient of the complex $Z \smallsetminus (R_1 \sqcup R_2)$ by identifying the two copies of $\partial R_1 \cong \partial R_2 \cong \Omega$ (see Figure 3). Performing this process for each periphery yields the complex A.

Figure 3. Part of a periphery P is shown on the right. The distinctly shaded region represents $K \cap P$ , where $K \subseteq Z$ is as above. The horizontal lines are contained in vertex spaces of Z. Outside of K, the horizontal lines do not intersect since all the vertices there are of degree 2. However, they may have a compact intersection inside K as shown in the figure. In this figure, the graph $\Omega \cong \partial R_1 \cong \partial R_2$ is a cycle with two finite trees hanging off it. On the left is part of a copy of $\mathbb{R} \times \Omega$ . The whole diagram represents an immersion $\mathbb{R} \times \Omega \longrightarrow P$ , which is an isomorphism outside of a compact set.

where the vertical maps are covering spaces, $S_0$ is the graph of spaces of a generalised Baumslag–Solitar group, and $T_0$ is its image in X. Covering maps are $\pi_1$ -injective and so is the map $S_0 \longrightarrow T_0$ by Lemma 4.4. Hence, $\mathbb{R} \times \Omega \longrightarrow P$ is $\pi_1$ -injective. Since $\partial R_i \hookrightarrow \mathbb{R} \times \Omega$ is a $\pi_1$ -isomorphism for $i = 1,2$ , it follows that the maps $\partial R_i \hookrightarrow P$ are $\pi_1$ -injective.

Proof. Denote the peripheries of Z by $P_1, \ldots, P_n$ and for each $i = 1, \ldots, n$ let $\Omega_i$ be the finite graph such that there is an immersion $\mathbb{R} \times \Omega_i \longrightarrow P_i$ . Since the peripheries are subgraphs of spaces of Z, the groups $\pi_1(\Omega_i) \cong \pi_1(\partial R_i)$ embed in $\pi_1(Z)$ by the previous claim. The quotient map induces an inclusion

\[ q_* \colon \pi_1(Z) \longrightarrow \pi_1(Z)*_{\pi_1(\Omega_1), \ldots, \pi_1(\Omega_n)} \cong \pi_1(A), \]

where $\pi_1(Z)*_{\pi_1(\Omega_1), \ldots, \pi_1(\Omega_n)}$ denotes the multiple HNN extension of $\pi_1(Z)$ over the subgroups $\pi_1(\Omega_i)$ .

The assumption that $\Phi_X$ is balanced and solvable implies that G does not contain any non-abelian generalised Baumslag–Solitar subgroups. Hence, $\pi_1(\Omega_i)$ is either trivial or isomorphic to $\mathbb{Z}$ . To see this, note that every periphery $P_i$ covers a generalised Baumslag–Solitar subcomplex $V_i \subseteq X$ , and therefore $\pi_1(V_i)$ cannot contain a non-abelian free subgroup. So from the fact that $\pi_1(\Omega_i) \leqslant \pi_1(V_i)$ we deduce that $\pi_1(\Omega_i)$ is either trivial or $\mathbb{Z}$ . The vertex group of a multiple HNN extension along trivial or infinite-cyclic subgroups is $L^2$ -independent by Lemma 3.8.

Step 4 ( $L^2$ -independence of vertex completion). As remarked in the previous step, A is a clean compact graph of graphs with $S^1$ edge spaces. Moreover, since the $\mathbb{Z}$ action on the strips was by covering translations, it follows that there is a natural quotient map $A \longrightarrow X$ whose restriction to every vertex space of A is an immersion. By adding edges to the vertex spaces of A, we can complete them to coverings of the corresponding vertex spaces of X. Denote the complex obtained from A in this way by B and note that $\pi_1(B) \cong \pi_1(A) * F$ , where F is a free group. Then $\pi_1(A)$ (and therefore H) is $L^2$ -independent in $\pi_1(B)$ .

Step 5 (Passing to finite-index and completing to a cover). In [Reference WiseWis00, Theorem 5.2, Steps 5 and 6], Wise shows how to pass to a finite-sheeted cover $B_1 \longrightarrow B$ which can be completed to a finite-sheeted cover $X_1 \longrightarrow X$ by attaching cylinders to $B_1$ .

The proof of Claim 4.17 shows that $\pi_1(A)$ has a graph of groups decomposition with underlying graph a rose, where the unique vertex group is H and the edge groups are either trivial or $\mathbb{Z}$ . Then $\pi_1(A) \cap \pi_1(B_1)$ also has a graph of groups decomposition with edge groups either trivial or $\mathbb{Z}$ , and $H \cap \pi_1(B_1)$ is a vertex group in this decomposition. By Lemma 3.7, $H \cap \pi_1(B_1)$ is $L^2$ -independent in $\pi_1(A) \cap \pi_1(B_1)$ .

As mentioned above, Wise shows that we can attach cylinders to $B_1$ to obtain a finite-sheeted covering $X_1 \longrightarrow X$ . Therefore $\pi_1(X_1)$ is a multiple HNN extension of $\pi_1(B_1)$ over cyclic subgroups, so that $\pi_1(B_1)$ (and thus $H \cap \pi_1(B_1)$ as well) is $L^2$ -independent in $\pi_1(X_1)$ . In summary, $H\cap \pi_1(B_1)$ has finite index in H and $H\cap \pi_1(B_1)$ is $L^2$ -independent in $\pi_1(X_1)$ , which has finite index in $\pi_1(X)$ . We conclude that G has the $L^2$ -Hall property by Corollary 3.20.

5.1 From graphs of free groups to limit groups

Before going into details of the work of Wilton, we first revisit Wise’s argument from § 4 to explain what are the main difficulties involved when dealing with ICE groups. We should remark that, in the context of limit groups, the process of getting virtually clean covers is hidden in the inductive argument and will not be mentioned again.

We denote by X a graph of spaces whose underlying graph has two vertices and one edge. We have that the vertex spaces are either graphs, as in Wise’s setting, or an ICE space Y and a torus $T^n$ , which is the case of interest in this section. The edge space of X is homeomorphic to a circle and the edge maps are assumed to be injective. Let H be a finitely generated subgroup of $\pi_1(X)$ and let $X_H\longrightarrow X$ be the covering corresponding to H. Scott’s criterion [Reference WiltonWil08, Lemma 1.3] topologically reformulates the subgroup separability of $\pi_1(X)$ as the ability to complete the precover $X_H$ of X to a finite-sheeted cover $\hat{X}\longrightarrow X$ so any prescribed finite subcomplex $\Delta$ of $X_H$ projects homeomorphically into $\hat{X}$ .

1. • The problem of ‘pruning’ in Wise’s argument corresponds to taking a core X’ of $X_H$ that contains $\Omega$ , the compact cores of the fundamental groups of each vertex space of $X_H$ and all the infinite-degree elevations of edge maps (i.e. the infinite strips). An important property of those elevations was that they escape any compact subset of the free splitting of the vertex space they belong to (because they act freely on the vertex graph). This property is called ‘properness’ (as introduced in [Reference WiltonWil08, Definition 2.12]) and is not satisfied by elliptic loops. Hence, in this setting the pruning must be performed more carefully. Conveniently, as a consequence of the 2-acylindricity of the Bass–Serre tree of an ICE group, non-elliptic (i.e. hyperbolic) loops are proper [Reference WiltonWil08, Lemma 2.16].

2. • The second part of Wise’s argument, which involves closing up the infinite strips, is another step towards obtaining a finite-sheeted precover from the precover X’ (since, after this, the preimages of points in the edge spaces are finite). However, one still has to figure out how to complete the precover so that preimages of points in the vertex spaces are finite as well. This corresponds to the problem of extending finite-sheeted covers of the edge spaces to finite-sheeted covers of the vertex spaces themselves. In this setting, this relies on a primitive version of omnipotence of free groups. This way, one constructs the precover $W \longrightarrow X'$ . Slightly more general conditions are offered in [Reference WiltonWil07, Section 3.2] in terms of homological assumptions on the edge spaces. However, for the case when Y is an ICE space and the edge subspace is generic, this problem is resolved in [Reference WiltonWil08] by strengthening the inductive hypothesis (incorporating the notion of tameness), so to have the required control on the prescribed collection of infinite-degree elevations.

3. • Lastly, the finite intermediate precover W is shown to admit a finite-sheeted covering $W_m\longrightarrow W$ that can be completed to a finite-sheeted covering $X_m\longrightarrow X$ . This is done similarly in Wilton’s argument when Y is an ICE space using the inductive hypothesis, without passing to a deeper $W_m$ .

The second point above explains Wilton’s observation [Reference WiltonWil08, Section 3] that the properties of local retractions of subgroup separability are not strong enough to serve as an induction hypothesis. We notice a similar problem. Following Scott’s philosophy, the $L^2$ -Hall property concerns the ability to complete precovers of X to finite-sheeted covers preserving the $L^2$ -homology in the process. The following example gives another reason for why this is not strong enough for an inductive argument either.

Example 5.4 illustrates that subgraphs of groups that are $L^2$ -injective on vertex groups need not be $L^2$ -injective overall, and so one needs some control of the non-trivial $L^2$ -classes that have non-trivial support on multiple vertex spaces. Wilton’s notion of tameness [Reference WiltonWil08, Definition 3.1] and Example 5.4 motivate the following notion of $L^2$ -tameness that allows us to inductively have such control.

1. (1) Every $\delta_j'$ descends to an elevation $\widehat{\delta}_j\colon \widehat{C_j}\longrightarrow \widehat X$ of degree d.

2. (2) The elevations $\widehat{\delta}_j$ are pairwise non-isomorphic.

3. (3) The subcomplex $\Delta$ injects into $\widehat{X}$ .

4. (4) The natural map $\pi_1(X')\longrightarrow \pi_1(\widehat X)$ extends to an injective and $L^2$ -injective map

   \[\pi_1(X')*\bigg(\coprod_{\mathcal L'}\mathbb{Z}\bigg)\longrightarrow\pi_1(\widehat{X}), \]
   defined as follows: if the copy of $\mathbb{Z}$ is labeled by $\delta_j'\in \mathcal L'$ then the element $1\in \mathbb{Z}$ is mapped to the class of the image of $ \widehat{\delta}_j'$ in $\pi_1(\widehat X)$ .

The subscripts i and j in Definition 5.5 are different, indicating that there may be several elevations $\delta_j'$ in $\mathcal L'$ for each $\delta_i$ in $\mathcal L$ .

As anticipated, the idea is that the strengthened version described in Definition 5.5 (with additional prescribed data relative to $\mathcal L$ ) admits a proof by induction and avoids bad embeddings like the one described in Example 5.4.

5.2 The proof of Theorem B

By the previous discussion, the following theorem implies Theorem B from the introduction, and its proof will occupy the remainder of this section.

Denote by $\{\delta_i \colon D_i \longrightarrow X\}$ and $\{\varepsilon_i \colon E_i \longrightarrow X\}$ the hyperbolic and elliptic loops of $\mathcal L$ respectively, relative to the splitting of X.

Step 1 (The precovers X’ and X”). Let $\Delta \subseteq X_H$ be a finite subcomplex and let $\big\{\delta_j^H\big\}$ and $\{\varepsilon_j'\}$ denote fixed sets of infinite-degree elevations of hyperbolic and elliptic loops, respectively, in $\mathcal L$ . We begin by taking a subcomplex $X'\subseteq X_H$ that satisfies the following conditions.

1. (1) The subcomplex X’ is a core for H, that is, X’ is a subgraph of spaces with finite underlying graph such that the induced map $\pi_1(X')\longrightarrow \pi_1(H)$ is an isomorphism.

2. (2) The subcomplex $\Delta$ is contained in X’.

3. (3) The image of each $\varepsilon_j'$ is contained in X’.

4. (4) Each infinite-degree elevation $\delta_j^H\colon \mathbb{R}\longrightarrow X_H$ restricts to a (possibly non-full) elevation $\delta_j'\colon D_j'\longrightarrow X'$ , where $D_j'\subseteq \mathbb{R}$ is a finite union of compact intervals.

The subscripts i and j are different, indicating that there may be several elevations $\delta_j^H$ for each $\delta_i$ , and likewise for $\varepsilon_i$ . We will keep completing the precover X’ further to get some more intermediate precovers $X'\subseteq X''\subseteq \overline{X}\subseteq X_H$ of X.

From [Reference WiltonWil08, Lemma 2.24], we can enlarge X’ to $X'' \subseteq X_H$ so that X” still enjoys properties (1)–(4) listed above while additionally satisfying that the corresponding elevations $\delta_j''\colon D_j''\longrightarrow X''$ are disparate (in the sense of [Reference WiltonWil08, Definition 2.2]). In particular, the induced map

(5.1) \begin{equation} \pi_1(X'') \longrightarrow \pi_1(X_H) \end{equation}

is an isomorphism.

Step 2 (The precover $\overline X$ ). We shall not need the definition of disparity but, instead, we will explain how this condition is used to extend the precover X” further. Recall that each $D_j''$ is the union of finitely many compact intervals and that $D_j''$ fits in the following commutative diagram.

For all sufficiently large positive integers d, there exists $\overline{D}_j\cong S^1$ such that $D_j'' \longrightarrow D_i$ factors through an embedding $D_j''\hookrightarrow \overline{D}_j$ and a d-sheeted covering map $\overline{D}_j\longrightarrow D_i$ . By [Reference WiltonWil08, Lemma 2.23], we can extend X” to a precover $\overline{X}$ such that each $\delta_j''$ extends to a full elevation $\overline{\delta}_j \colon \overline{D}_j \longrightarrow \overline{X}$ and the diagram

commutes. By possibly enlarging $\Delta$ , we can assume that the images of the $\overline{\delta}_j$ are contained in $\Delta$ .

In the construction of [Reference WiltonWil08, Lemma 2.23], one first enlarges X” to X”’ by adding some simply connected vertex spaces of $X_H$ to obtain $X'' \hookrightarrow X''' \hookrightarrow X_H$ . In particular, the induced map $\pi_1(X'')\longrightarrow \pi_1(X''')$ is an isomorphism. Then one considers a collection of pairs $(\phi_{k, \mathsf{o}}^H, \phi_{k, \mathsf{t}}^H)$ of edge maps $\phi_{k, \mathsf{o}}^H\colon \mathbb{R}_{\mathsf{o}}\longrightarrow X_H$ and $\phi_{k, \mathsf{t}}^H\colon \mathbb{R}_{\mathsf{t}}\longrightarrow X_H$ which are elevations of the incident and terminal edge maps of some edge space $S_k^1$ of X. Furthermore, these pairs $(\phi_{k, \mathsf{o}}^H, \phi_{k, \mathsf{t}}^H)$ will have the property that these are not edge maps of X”’ (such elevations are usually called hanging elevations of the precover X”’, as in [Reference WiltonWil08, Remark 2.18]). For each k, we denote by $\mathbb{R}_{\mathsf{o}}$ and $\mathbb{R}_{\mathsf{t}}$ the domains of $\phi_{k, \mathsf{o}}^H$ and $\phi_{k, \mathsf{t}}^H$ , respectively (which are the universal covers of $S_k^1$ ). We fix a deck transformation $\tau\colon \mathbb{R}_{\mathsf{o}}\longrightarrow \mathbb{R}_{\mathsf{t}}$ so that the natural diagram