Uncountably many Poincare duality groups of type FP

July 10, 2026

Uncountably many Poincare duality groups of type FP

Finiteness properties of groups have long been a topic of great interest to group theorists. For instance, Higman's celebrated embedding theorem says that a finitely generated group is a subgroup of a finitely presented group if and only if it is recursively presented. Asking what sort of finiteness properties subgroups of type $F$ (i.e. finite classifying space) have is also an active area of research: it was only recently shown that the finiteness properties are all distinguished within the class of subgroups of hyperbolic groups (torsion-free hyperbolic groups are of type $F$). Before that, there were homological and homotopical finiteness properties of groups, or equivalently, their classifying spaces, and a major achievement of Bestvina-Brady was to show that these are very different using combinatorial Morse theory. In particular, given an acyclic but not contractible finite flag complex $L$, the associated Bestvina-Brady group $BB_L$ is not finitely presented, but is of type $FP(\mathbb{Z})$. It is then natural to wonder how big the difference between finitely presented and $FP(\mathbb{Z})$ is. Leary showed that it is huge: there are uncountably many Poincare duality groups of type $FP(\mathbb{Z})$ which aren't finitely presented, and later upgraded this to uncountably many QI-classes. Compare this with the fact that there are only countably many finite presentations. Here I'll say a little about the constructions, and encourage the interested reader to take a look at the original papers, which are quite well-written and contain computations of a number of interesting cohomological invariants. I'll assume some amount of familiarity with the Bestvina-Brady construction, which I wrote up here.

As a reminder about conventions, each graph $\Gamma$ that we consider will be connected, simplicial, and will sometimes be viewed as a metric space via the path metric $d_\Gamma$, in which each edge has length one. The induced metric on the vertex set of a Cayley graph $\Gamma(G,S)$ is thus the $S$-word length metric on the group $G$.

We denote by $X_L$ the universal cover of the Salvetti complex of the RAAG over a complex $L$. Leary interprets these as part of a family indexed by subsets of $\mathbb{Z}$ and constructs the uncountably many groups as groups of deck transformations of some regular branched covering $ X_L^{(S)} \to X_L/\mathrm{BB}_L $. (A workhorse way of getting uncountably many objects is get one for every subset of $\mathbb{N}$. As another aside, branched covers are surprisingly good at producing interesting objects: any 3-manifold arises as a branched cover over the sphere, Riemann surfaces branching over each other produces a very rich theory, Gromov-Thurston manifolds are branched covers (see also this previous post)...)

Defining the complexes

We first recall a bit more about Salvetti complexes. The link of the point 0 in the circle $\mathbb{R}/\mathbb{Z}$ is a copy of the 0-sphere, corresponding to travelling in the positive and negative directions along $\mathbb{R}$. Similarly, the link of the 0-cell in $T_v$ is a copy of the 0-sphere, which we shall denote by $\mathbb{S}(v) = \{v^+, v^-\}$. Under the isomorphism of links induced by the homeomorphism $l_v : T_v \to \mathbb{R}/\mathbb{Z}$, the point $v^+$ (resp. $v^-$) corresponds to the positive direction (resp. negative direction). The link of the vertex in $T_\Delta$ is the $(|L^0| - 1)$-sphere $\mathbb{S}(\Delta)$, equal to the join $\mathbb{S}(\Delta) = *_{v \in \Delta} \mathbb{S}(v)$. For each simplex $\sigma$ of $L$, there is an inclusion $\mathbb{S}(\sigma) \subseteq \mathbb{S}(\Delta)$, and the link $\mathbb{S}(L)$ of the unique vertex of $T_L$ is equal to the union of these images:
\[
\mathbb{S}(L) = \bigcup_{\sigma \in L} \mathbb{S}(\sigma) \subseteq \mathbb{S}(\Delta).
\]
We will use this description, and an additional property of simplicial complexes that we now define.

Proposition 1: For a simplicial complex $ L $ without isolated vertices, the following properties are equivalent:

Every 1-simplex is contained in a 2-simplex and every vertex link is connected;
The topological realization $|L|$ has no local cut point.

Proof: For $ n \geq 2 $ the $ n $-simplex has no local cut points; it follows that the only possible local cut points are either vertices of $ L $ or points in an edge of $ L $ that is not contained in any triangle. A non-isolated vertex is a local cut point if and only if its link is not connected, and the midpoint of an isolated edge is always a local cut point. $\blacksquare$

Definition: Say that $ L $ has nlcp if it satisfies the equivalent conditions listed above.

A throwaway remark: having no local cut points in the boundary of a hyperbolic group says that the group is rigid by work of Bowditch, but it seems unclear to me if there is a deeper connection.

Lemma 2: For any flag complex $ L $ without isolated vertices, there is a flag complex $ M = M(L) $ with $ L \subseteq M $, and having the following properties.

$ M $ has nlcp;
$ L $ is a full subcomplex of $ M $;
The inclusion $ L \subseteq M $ is a homotopy equivalence;
Provided that $ L $ is at least 2-dimensional, $ M $ has the same dimension as $ L $.

The construction $ L \mapsto M(L) $ can be chosen to be functorial for maps $ L_1 \to L_2 $ that do not collapse any simplex.

Any triangulation of $ |L| \times [0, 1] $ for which $ |L| \times \{0\} $ is realized as a full subcomplex isomorphic to $ L $ will have properties (1)-(3). For property (4) to hold as well, Leary takes $ M(L) $ to be a suitable triangulation of the mapping cylinder of the inclusion of the 1-skeleton $ L^1 $ in $ L $.

The reason we care about nlcp is the following auxiliary result, which we need later:

Proposition 3: $L$ is a retract of $\mathbb{S}(L)$, so the natural map $\pi_1(\mathbb{S}(L)) \to \pi_1(L)$ is always surjective. If $ L $ has nlcp, then this map is an isomorphism.

Now here is the construction:

Theorem 4: For any connected finite $ L $ and any $ S \subseteq \mathbb{Z} $, the locally CAT(0) cubical complex $ X_L / \mathrm{BB}_L $ admits a regular branched cover $ c : X_L^{(S)} \to X_L / \mathrm{BB}_L $ with the following properties:

$ c $ factors through $ X_L $: there is a regular branched cover $ b : X_L^{(S)} \to X_L $ whose composite with the natural map $ X_L \to X_L / \mathrm{BB}_L $ is equal to $ c $;
The function $ f^{(S)} := f \circ b = \tilde{l} \circ c $ is a Morse function on $ X^{(S)} $;
The links of vertices of $ X^{(S)} $ are either $ \mathbb{S}(L) $ or $ \mathbb{S}(\tilde{L}) $, depending on their height:\[\mathrm{Lk}_{X^{(S)}}(v) =\begin{cases}\mathbb{S}(L) & \text{if } f^{(S)}(v) \in S \\\mathbb{S}(\tilde{L}) & \text{if } f^{(S)}(v) \notin S\end{cases}\]
The ascending and descending links of vertices of $X^{(S)}$ are either $L$ or $\tilde{L}$ depending on their height:

\[
\operatorname{Lk}|_{X^{(S)}}(v) = \operatorname{Lk}\!\uparrow_{X^{(S)}}(v) =
\begin{cases}
L & \text{if } f^{(S)}(v) \in S \\
\tilde{L} & \text{if } f^{(S)}(v) \notin S
\end{cases}
\]
The group $G_L(S)$ is by definition the group of deck transformations of the branched covering $c : X_L^{(S)} \to X_L / \mathrm{BB}_L$.

We will just give the construction of the groups and omit the proofs of the later properties. To justify the construction, we need a result on cube complexes, that may be of independent interest, about how branching works.

Theorem 5: Let $X$ be a locally CAT(0) cube complex with universal covering $p : \tilde{X} \to X$, suppose that all vertex links in $X$ are connected, and let $V$ be a set of vertices of $X$.
There is a CAT(0) cubical complex $\overline{X}$, which turns out to be the metric completion of $Y$ in the path metric induced by $c : Y \to X \setminus V$ and a regular branched covering map $c : \overline{X} \to X$, with branching locus contained in V and such that

\[
\mathrm{Lk}_{\overline X}(w) =
\begin{cases}
\mathrm{Lk}_X(c(w)) & \text{if } c(w) \notin V \\
\widetilde{\mathrm{Lk}_X(c(w))}& \text{if } f^{(S)}(v) \in V
\end{cases}
\]

Moreover, the covering map $c$ factors through $\tilde{X} $, in the sense that $c = p \circ b$ for some regular branched covering map $b : \overline X \to \tilde{X} $.

Leary gives the combinatorial construction first and adds later that it is a completion, but I think it's a bit easier to follow what's going on and see how he came up with it knowing that it's meant to be some sort of completion. Anyway, let's see how to use this.

Construction for theorem 4: In the case when $L$ has nlcp, Proposition 3 shows that $\mathbb{S}(\tilde{L})$ is the universal cover of $\mathbb{S}(L)$; in this case $X_L^{(S)}$ and $b$ can be constructed by applying Theorem 5 to the locally CAT(0) cube complex $X_L / \mathrm{BB}_L$, with $V$ equal to the vertices of $X_L / \mathrm{BB}_L$ whose height lies in $\mathbb{Z} - S$. Since $X_L / \mathrm{BB}_L$ has exactly one vertex of each height we note that $G_L(S)$ acts transitively on the vertices of $X_L^{(S)}$ of each height.

In the general case, we realize $X_L^{(S)}$ as a subcomplex of $X_M^{(S)}$, where $M = M(L)$ is as described in Lemma 2. We have $X_L / A_L$ embedded as a subcomplex of $X_M / A_M$. Define $Y$ to be one of the connected components of the inverse image of $X_L / A_L \subseteq X_M / A_M$ under the composite map $X_M^{(S)} \to X_M \to X_M / A_M$. Since the group $G_M(S)$ acts transitively on the vertices of $X_M^{(S)}$ of a given height and $Y$ contains vertices of all heights, any other connected component will be isomorphic to $Y$. Since $L$ is a full subcomplex of $M$, $\mathbb{S}(L)$ is a full subcomplex of $\mathbb{S}(M)$, and the inverse image of $\mathbb{S}(L) \subseteq \mathbb{S}(M)$ under the universal covering map $\mathbb{S}(\widetilde{M}) \to \mathbb{S}(M)$ is $\mathbb{S}(\tilde{L})$. It follows that vertices of $Y$ of height in $S$ have link $\mathbb{S}(L)$ and vertices of $Y$ of height not in $S$ have link $\mathbb{S}(\tilde{L})$ as required.

It turns out that these groups admit the following alternative descriptions. In particular, we will see a completely explicit presentations that realise the groups $ G_L(S) $ and the homomorphisms $ G_L(S) \twoheadrightarrow G_L(T) $ for $ 0 \in S \subseteq T $, in the sense that the generating set depends only on $ L $ while the relators in $ P_L(\Gamma, S) $ are a subset of the relators in $ P_L(\Gamma, T) $. Since there are isomorphisms $ G_L(S) \cong G_L(T) $ whenever $ T = S + n $ is a translate of $ S $, the presentations $ P_L(\Gamma, S) $ describe the isomorphism type of each $ G_L(S) $ except for $ G_L(\emptyset) $, which can be described as a semidirect product $ \mathrm{BB}_{\widetilde{L}} \rtimes \pi_1(L) $.

Recall that a directed edge in a simplicial complex is an ordered pair $ (v, v') $ of vertices such that the corresponding unordered pair is an edge. A directed loop $ \gamma = (a_1, \ldots, a_l) $ of length $ l $ is an ordered $ l $-tuple of directed edges whose endpoints match up, in the sense that there are vertices $ v_0, \ldots, v_l $ with $ v_0 = v_l $ and $ a_i = (v_{i-1}, v_i) $ for each $ i $. In the following definition, we assume that $ L $ is a finite connected flag complex, and that $ \Gamma $ is a finite collection of directed edge loops in $ L $ that normally generates $ \pi_1(L) $; equivalently if one attaches discs to $ L $ along the loops in $ \Gamma $ one obtains a simply-connected complex.

Definition: For $ L $ and $ \Gamma $ as above and for any set $ S $ with $ 0 \in S \subseteq \mathbb{Z} $, the presentation $ P_L(\Gamma, S) $ has as generators the directed edges of $ L $, subject to the following relations.

(Edge relations.) For each directed edge $ a = (x, y) $ with opposite edge $ \overline{a} = (y, x) $, the relation $ a\overline{a} = 1 $;
(Triangle relations.) For each directed triangle $ (a, b, c) $ in $ L $, the relations $ abc = 1 $ and $ a^{-1}b^{-1}c^{-1} = 1 $;
(Long cycle relations.) For each $ n \in S - \{0\} $, and each $ (a_1, a_2, \ldots, a_l) \in \Gamma $, the relation $ a_1^n a_2^n \cdots a_l^n = 1 $.

In the case when $ L $ is simply connected, we may take $ \Gamma $ to be empty, in which case $ P_L(\Gamma, S) $ is independent of $ S $. This reflects the fact that when $ L $ is simply-connected, the natural map $ G_L(S) \to G_L(\mathbb{Z}) \cong \mathrm{BB}_L $ is an isomorphism for each $ S $.

Thus, the $G_L(S)$ have the following descriptions. They are:

the group of deck transformations of the regular branched covering $ X_L^{(S)} \to X_L/\mathrm{BB}_L $;
the group of deck transformations of the regular covering $ X_t^{(S)} \to X_t/\mathrm{BB}_L $, where $ X_t^{(S)} $ denotes a non-integer level set in $ X_L^{(S)} $ and similarly $ X_t $ in $ X_L $;
the fundamental group of a space obtained from $ X_L/\mathrm{BB}_L $ by removing some vertices, in the case when $ L $ has nlcp;
the group given by a presentation $ P_L(\Gamma, S) $, in the case when $ 0 \in S $.

By using discrete Morse theory analogous to the Bestvina-Brady paper, Leary proves the following result on their finiteness properties:

Theorem 6: Let $ L $ be a connected finite flag complex, let $ R $ be any ring in which $ 1 \neq 0 $, and let $ S_0 $ be any subset of $ \mathbb{Z} $ such that both $ S_0 $ and $ \mathbb{Z} - S_0 $ are infinite. The following are equivalent:

$ L $ and $ \widetilde{L} $ are $ R $-acyclic;
For each $ S \subseteq \mathbb{Z} $, $ G_L(S) $ is type $ FH(R) $;
For each $ S \subseteq \mathbb{Z} $, $ G_L(S) $ is type $ FP(R) $;
$ \pi_1(L) $ and $ G_L(S_0) $ are type $ FP(R) $.

For each $ n \geq 2 $, the following are equivalent:

$ L $ and $ \widetilde{L} $ are $(n-1)$-$ R $-acyclic;
For each $ S \subseteq \mathbb{Z} $, $ G_L(S) $ is type $ FH_n(R) $;
For each $ S \subseteq \mathbb{Z} $, $ G_L(S) $ is type $ FP_n(R) $;
$ \pi_1(L) $ and $ G_L(S_0) $ are type $ FP_n(R) $.

Quasi-isometry classes

I have yet to say anything about the promised uncountably many QI classes since, for all we know right now, we could somehow end up with the same group for many choices of $S$. The tool for distinguishing groups up to QI is due to Bowditch, and we follow the exposition of R.Kropholler--Leary--Soroko.

For $k > 0$ an integer, recall that a function $f: X \to Y$ between metric spaces is \emph{$k$-Lipschitz} if $d_Y(f(x), f(x')) \leq k \cdot d_X(x, x')$ for all $x, x' \in X$. Following Bowditch, say that graphs $\Gamma$ and $\Lambda$ are \emph{$k$-quasi-isometric} if there exist a pair of $k$-Lipschitz maps of vertex sets $\phi: V(\Gamma) \to V(\Lambda)$ and $\psi: V(\Lambda) \to V(\Gamma)$ so that $d_\Gamma(x, \psi \circ \phi(x)) \leq k$ for each vertex $x$ of $\Gamma$ and similarly $d_\Lambda(y, \phi \circ \psi(y)) \leq k$ for each vertex $y$ of $\Lambda$. Graphs are \emph{quasi-isometric} if they are $k$-quasi-isometric for some integer $k > 0$.

(This isn't the standard definition, but it is true that graphs $\Gamma$, $\Lambda$ are quasi-isometric as above if and only if the metric spaces $(\Gamma, d_\Gamma)$ and $(\Lambda, d_\Lambda)$ are quasi-isometric in the usual sense.)

An edge loop of length $l$ in a (simplicial) graph $\Gamma$ is a sequence $v_0, \ldots, v_l$ of vertices such that $v_0 = v_l$ and $\{v_{i-1}, v_i\}$ is an edge for $1 \leq i \leq l$. For a graph $\Gamma$ and an integer constant $l$, let $\Gamma_l$ denote the 2-complex whose 1-skeleton is the geometric realization of $\Gamma$, with one 2-cell attached to each edge loop in $\Gamma$ of length strictly less than $l$. An edge loop of length $l$ in $\Gamma$ is said to be \emph{taut} if it is not null-homotopic in $\Gamma_l$. Bowditch's \emph{taut loop length spectrum} $H(\Gamma)$ for the graph $\Gamma$ is the set of lengths of taut loops.

Bowditch defines subsets $H, H' \subseteq \mathbb{N}$ to be $k$-related if for all $l > k^2 + 2k + 2$, whenever $l \in H$ then there is some $l' \in H'$ with $l/k < l' < lk$ and vice-versa. He then proves:

Lemma 7: If (connected) graphs $\Gamma$ and $\Lambda$ are $k$-quasi-isometric, then $H(\Gamma)$ and $H(\Lambda)$ are $k$-related.

R.Kropholler--Leary--Soroko estimate the set $H(\Gamma(S))$, where $\Gamma(S)$ is the Cayley graph associated to the natural generating set for $G_L(S)$. The natural presentation for $G_L(S)$ contains relators whose lengths are parametrized by the absolute values of the members of $S$, but it also contains many relators of length 3, and does not satisfy the $C'(1/6)$ condition. To apply Bowditch's technique we need a lower bound for the word lengths of elements in the kernel of the map $G_L(S) \to G_L(T)$ for $S \subseteq T$, in terms of $T - S$. The Cayley graph $\Gamma(S)$ embeds naturally in a CAT(0) cubical complex.

Define the singular set for a map consists of all points at which it is not a local isometry. When I spoke to Ian about these results, he said that the following lemma, which is in the spirit of estimating various distances, was a result they were very pleased by:

Lemma 8: Let $f : X \to Y$ be a continuous map of CAT(0) metric spaces, and suppose that $x \neq x'$ but $f(x) = f(x')$. Then the distance $d_X(x,x')$ is at least the sum of the distances from $x$ and $x'$ to the singular set for $f$.

QI classes of the $G_L(S)$

Fix a finite connected non-simply connected flag complex $L$. For $S$ a subset of $\mathbb{Z}$ containing $0$, let $\Gamma(S)$ denote the Cayley graph of $G_L(S)$ with respect to the standard generators. It will be convenient to assume that elements of $S$ grow quickly, which we do as follows.

Define $\alpha = \alpha(L)$ by $\alpha = \sqrt{2/(d+1)}$, where $d$ is the dimension of $L$. For a finite set $\Omega$ of loops in $L$ that normally generates $\pi_1(L)$, let $\beta(L,\Omega)$ be the maximum of the lengths of the loops in $\Omega$, and define $\beta = \beta(L)$ to be the minimum value of $\beta(L,\Omega)$ over all such $\Omega$. Choose an integer constant $C = C(L)$ so that $C > \beta/\alpha$ and $C\alpha > 3$. For $F$ any subset of $\mathbb{N}$, define $S(F) = \{0\} \cup \{C^{2^n} : n \in F\}$. By estimating the taut length spectrum, the authors prove an analogue of Bowditch's result:

Proposition 9: If $F, F'$ are subsets of $\mathbb{N}$ so that $\Gamma(S(F))$ and $\Gamma(S(F'))$ are quasi-isometric then the symmetric difference of $F$ and $F'$ is finite.

From this it follows that for each fixed finite connected flag complex $L$ that is not simply-connected, there are continuously many quasi-isometry classes of groups $G_L(S)$.

Poincare duality

We now discuss the long promised Poincare duality. Recall that a group is said to be a Poincare duality group, or a PD$^n$ group, if it is of type $FP_{\infty}$, has $H^n(G, \mathbb{Z})=\mathbb{Z}$, and cup product induces a perfect pairing between $H^i$ and $H^{n-i}$. These are satisfied by the cohomology of a closed manifold of dimension $n$, and it is a very interesting question whether Poincare duality groups must come from a manifold. Davis was the first to show that a hypothesis of finite presentability is necessary, using his famous reflection group trick. It remains, to the best of my knowledge, open whether every finitely presented Poincare duality group is a manifold group. We will see how to use the above to produce uncountably many Poincare duality groups.

Suppose that a group $G$ acts by automorphisms on the graph $K$. This induces an action of $G$ on $W_K$ by automorphisms, permuting the given generators for $W_K$, and so there is a semidirect product group $J = W_K \rtimes G$. Identify $G$ with its image inside the semidirect product $J$. A choice of generating set for $G$ together with a choice of $G$-orbit representatives in $V(K)$ gives rise to a generating set for $J$.

Now suppose that $S \mapsto G(S)$ is a functor from the category of subsets of $\mathbb{Z}$ with inclusions as morphisms to the category of finitely generated groups and surjective homomorphisms; for example $S \mapsto G_L(S)$ is such a functor for any connected finite flag complex $L$. Suppose further that $G(\emptyset)$ acts freely cocompactly on a (simplicial) graph $K(\emptyset)$ in such a way that any two vertices in the same $G(\emptyset)$-orbit are at edge path distance at least four. For $S \subseteq \mathbb{Z}$, define $K(S)$ to be the quotient of $K(\emptyset)$ by the kernel of the map $G(\emptyset) \to G(S)$, so that $G(S)$ acts freely cocompactly on the graph $K(S)$.

For $S \subseteq \mathbb{Z}$, define $J(S)$ to be the semidirect product $W_{K(S)} \rtimes G(S)$. Then $S \mapsto J(S)$ is another functor from subsets of $\mathbb{Z}$ and inclusions to finitely generated groups and surjective group homomorphisms. Fix a finite generating set for $J(\emptyset)$ consisting of a finite generating set for $G(\emptyset)$ and a set $V'$ of $G(\emptyset)$-orbit representatives in $V(K(\emptyset))$. As generating set for $J(S)$, take the image of our given generating set for $J(\emptyset)$, and as generating set for $G(S)$ take the image of our given generating set for $G(\emptyset)$. For each $S$, the generating set for $G(S)$ is a subset of the generating set for $J(S)$, and its complement consists of generators that are in the kernel of the map $J(S) \to G(S)$.

In this case, R.Kropholler--Leary--Soroko prove analogous results for the semidirect product $J(S) := W_{K(S)} \rtimes G_L(S)$. Define constants $\alpha = \alpha(L)$, $\beta := \beta(L)$ and $C := C(L)$ as in the previous section, and for $F \subseteq \mathbb{N}$define $S(F) \subseteq \mathbb{N}$ as before, and denote by $\Lambda(S(F))$ the Cayley graph $\Gamma(J(S(F)))$. The analogous results are:

Lemma 10: If $F$, $F'$ are subsets of $\mathbb{N}$ so that $\Lambda(S(F))$ and $\Lambda(S(F'))$ are quasi-isometric, the symmetric difference of $F$ and $F'$ is finite.

Lemma 11: For any $L$ that is not simply-connected, and any graph $K(\emptyset)$ with a free $G_L(\emptyset)$-action, there are continuously many quasi-isometry classes of groups $J(S)$.

The reason why Lemma 11 is of value concerns the use of the Davis trick to construct non-finitely presented Poincaré duality groups. The starting point is a $2$-complex $L$ for which each $G_L(S)$ is type $FP$; for this group there is a finite $2$-complex that is an Eilenberg-Mac Lane space $K(G_L(\emptyset),1)$. For any $n \geq 4$, one can find a compact $n$-manifold $V$ with boundary that is also a $K(G_L(\emptyset),1)$. Now let $K$ be the $1$-skeleton of the barycentric subdivision of a triangulation of the boundary of $V$, and let $K(\emptyset)$ be the $1$-skeleton of the induced triangulation of the boundary of the universal cover of $V$, with $G_L(\emptyset)$ acting via deck transformations. For this choice of $K(\emptyset)$, the group $J(\emptyset)$ contains a finite-index torsion-free subgroup $J'$ that is the fundamental group of a closed aspherical $n$-manifold $M$, and such that regular covering $M(S)$ of $M$ with fundamental group the kernel of $J' \to J(S)$ is acyclic for each $S \subseteq \mathbb{Z}$. One deduces that each $J(S)$ contains a finite-index torsion-free subgroup that is a Poincaré duality group of dimension $n$. Since the inclusion of a finite-index subgroup is always a quasi-isometry, Lemma 11 implies

Corollary 12: For each $ n \geq 4 $ there are continuously many quasi-isometry classes of non-finitely presented $ n $-dimensional Poincaré duality groups.

By the Schwarz-Milnor Lemma, the acyclic covering manifold $M(S)$ of $M$ is quasi-isometric to the group $J(S)$, so we get

Corollary 13: For each $ n \geq 4 $ there is a closed aspherical $ n $-manifold admitting continuously many quasi-isometry classes of regular acyclic covers.

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On the shoulders of giants

Uncountably many Poincare duality groups of type FP

Defining the complexes

Quasi-isometry classes

QI classes of the \(G_L(S)\)

Poincare duality

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