Quasi-isometry Classification Research Articles

Continuously many quasi-isometry classes of residually finite groups

AbstractWe study a family of finitely generated residually finite small-cancellation groups. These groups are quotients of $F_2$ depending on a subset $S$ of positive integers. Varying $S$ yields continuously many groups up to quasi-isometry.

Measure equivalence classification of transvection-free right-angled Artin groups

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group G is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when G is nonabelian, the automorphism group of the universal cover of the Salvetti complex of G always contains infinitely generated (non-uniform) lattices.

Quasi‐isometric diversity of marked groups

We use basic tools of descriptive set theory to prove that a closed set $\mathcal S$ of marked groups has $2^{\aleph_0}$ quasi-isometry classes provided every non-empty open subset of $\mathcal S$ contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains $2^{\aleph_0}$ quasi-isometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. They can also be used to prove the existence of $2^{\aleph_0}$ quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties.

Quasiflats in hierarchically hyperbolic spaces

The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Several noteworthy examples for which the rank coincides with familiar quantities include: the dimension of maximal Dehn twist flats for mapping class groups; the maximal rank of a free abelian subgroup for right-angled Coxeter groups and right-angled Artin groups (in the latter this can also be observed as the clique number of the defining graph); and, for the Weil–Petersson metric, the rank is the integer part of half the complex dimension of Teichmüller space. We prove that, in a hierarchically hyperbolic space (HHS), any quasiflat of dimension equal to the rank lies within finite distance of a union of standard orthants (under a very mild condition on the HHS satisfied by all natural examples). This resolves outstanding conjectures when applied to a number of different groups and spaces. In the case of the mapping class group, we verify a conjecture of Farb. For Teichmüller space we answer a question of Brock. In the context of certain CAT(0) cubical groups, our result handles novel special cases, including right-angled Coxeter groups. An important ingredient in the proof, which we expect will have other applications, is that the hull of any finite set in an HHS is quasi-isometric to a CAT(0) cube complex of dimension bounded by the rank. (If the HHS is a CAT(0) cube complex, then the rank can be lower than the dimension of the space.) We deduce a number of applications of these results. For instance, we show that any quasi-isometry between HHSs induces a quasi-isometry between certain factored spaces, which are simpler HHSs. This allows one, for example, to distinguish quasi-isometry classes of right-angled Artin/Coxeter groups. Another application of our results is to quasi-isometric rigidity. Our tools in many cases allow one to reduce the problem of quasi-isometric rigidity for a given hierarchically hyperbolic group to a combinatorial problem. As a template, we give a new proof of quasi-isometric rigidity of mapping class groups, which, once we have established our general quasiflats theorem, uses simpler combinatorial arguments than in previous proofs.

Divergence and quasi-isometry classes of random Gromov’s monsters

AbstractWe show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders.Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.

Cohomological induction and uniform measure equivalence

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.

Quasi-isometry classification of right-angled Artin groups that split over cyclic subgroups

For a one-ended right-angled Artin group, we give an explicit description of its JSJ tree of cylinders over infinite cyclic subgroups in terms of its defining graph. This is then used to classify certain right-angled Artin groups up to quasi-isometry. In particular, we show that if two right-angled Artin groups are quasi-isometric, then their JSJ trees of cylinders are weakly equivalent. Although the converse to this is not generally true, we define quasi-isometry invariants known as stretch factors that can distinguish quasi-isometry classes of RAAGs with weakly equivalence JSJ trees of cylinders. We then show that for many right-angled Artin groups, being weakly equivalent and having matching stretch factors is a complete quasi-isometry invariant.

Actions of small cancellation groups on hyperbolic spaces

We generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set \({\mathcal {TC}}\) of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases \({\mathcal {TC}}\) is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions \([G\curvearrowright X] \preceq [G\curvearrowright Y]\) in this poset, there is an embeddeding \(\iota :P(\omega )\rightarrow {\mathcal {TC}}\) such that \(\iota (\emptyset )=[G\curvearrowright X]\) and \(\iota ({\mathbb {N}})=[G\curvearrowright Y]\). We use this poset to prove that there are uncountably many quasi-isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both.

Surface group amalgams that (don't) act on 3-manifolds

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of $3$-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a $3$-manifold. We prove that each such surface group amalgam is abstractly commensurable to a right-angled Coxeter group from a related family. In an appendix, we determine the quasi-isometry classes among these surface amalgams and their related right-angled Coxeter groups.

On the coarse geometry of certain right-angled Coxeter groups

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex torsion free infinite index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.

Embedding certain diffeomorphism groups in the quasi-isometry groups of Euclidean spaces

We show that certain groups of diffeomorphisms and PL-homeomorphisms embed in the group of all quasi-isometry classes of the Euclidean spaces.

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical \mathrm {Gr}'(1/6) small cancellation groups. In particular, we characterize their ‘contracting geodesics,’ which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group G containing an element g that is strongly contracting with respect to one finite generating set of G and not strongly contracting with respect to another. In the case of classical C'(1/6) small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical \mathrm {Gr}'(1/6) small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical \mathrm {Gr}'(1/6) small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.

On positive scalar curvature and moduli of curves

In this article we first show that any finite cover of the moduli space of closed Riemann surfaces of genus $g$ with $g\geq 2$ does not admit any Riemannian metric $ds^2$ of nonnegative scalar curvature such that $ds^2 \succ ds_{T}^2$ where $ds_{T}^2$ is the Teichm\"uller metric. Our second result is the proof that any cover $M$ of the moduli space $\mathbb{M}_{g}$ of a closed Riemann surface $S_{g}$ does not admit any complete Riemannian metric of uniformly positive scalar curvature in the quasi-isometry class of the Teichm\"uller metric, which implies a conjecture of Farb-Weinberger.

Detecting a subclass of torsion-generated groups

We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurable to a group generated by finite-order elements. Consequently, the subclass of groups in this class generated by finite-order elements is not quasi-isometrically rigid. We provide necessary conditions for two groups in this class to be abstractly commensurable. We use these conditions to prove there are infinitely many abstract commensurability classes within each quasi-isometry class within this class that contains a group generated by finite-order elements.

A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group’s defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group’s order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n > 1 , we give examples of right-angled Coxeter groups which are thick of order n , yet are algebraically thick of order strictly larger than n, answering a question of Behrstock-Druţu-Mosher.

Simple groups separated by finiteness properties

We show that for every positive integer n there exists a simple group that is of type $$\mathrm {F}_{n-1}$$ but not of type $$\mathrm {F}_n$$ . For $$n\ge 3$$ these groups are the first known examples of this kind. They also provide infinitely many quasi-isometry classes of finitely presented simple groups. The only previously known infinite family of such classes, due to Caprace–Remy, consists of non-affine Kac–Moody groups over finite fields. Our examples arise from Rover–Nekrashevych groups, and contain free abelian groups of infinite rank.

Bowditch's JSJ tree and the quasi‐isometry classification of certain Coxeter groups

Bowditch's JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian. Our main result gives an explicit, computable "visual" construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the cocompact Fuchsian groups form a rigid quasi-isometry class. In an appendix, written jointly with Christopher Cashen, we show that the JSJ tree is not a complete quasi-isometry invariant for the entire class of Coxeter groups we consider.

Uncountably many quasi-isometry classes of groups of type FP

Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincar\'e duality groups for each $n\geq 4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.

On the large-scale geometry of diffeomorphism groups of 1-manifolds

Abstract We apply the framework of Rosendal to study the large-scale geometry of the topological groups Diff + k ⁡ ( M 1 ) {\operatorname{Diff}_{+}^{k}(M^{1})} , consisting of orientation-preserving C k {C^{k}} -diffeomorphisms (for 1 ≤ k ≤ ∞ {1\leq k\leq\infty} ) of a compact 1-manifold M 1 {M^{1}} ( = I {=I} or 𝕊 1 {\mathbb{S}^{1}} ). We characterize the relative property (OB) in such groups: A ⊆ Diff + k ⁡ ( M 1 ) {A\subseteq\operatorname{Diff}_{+}^{k}(M^{1})} has property (OB) relative to Diff + k ⁡ ( M 1 ) {\operatorname{Diff}_{+}^{k}(M^{1})} if and only if sup f ∈ A sup x ∈ M 1 | log f ′ ( x ) | < ∞ {\sup_{f\in A}\sup_{x\in M^{1}}\lvert\log f^{\prime}(x)|<\infty} and sup f ∈ A ⁡ sup x ∈ M 1 ⁡ | f ( j ) ⁢ ( x ) | < ∞ {\sup_{f\in A}\sup_{x\in M^{1}}|f^{(j)}(x)|<\infty} for every integer j with 2 ≤ j ≤ k {2\leq j\leq k} . We deduce that Diff + k ⁡ ( M 1 ) {\operatorname{Diff}_{+}^{k}(M^{1})} has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if k < ∞ {k<\infty} . We show that the groups Diff + 1 ⁡ ( I ) {\operatorname{Diff}_{+}^{1}(I)} and Diff + 1 ⁡ ( 𝕊 1 ) {\operatorname{Diff}_{+}^{1}(\mathbb{S}^{1})} are quasi-isometric to the infinite-dimensional Banach space C ⁢ [ 0 , 1 ] {C[0,1]} .

Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams