# Questions tagged [geometric-group-theory]

Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

651
questions

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### $V$-like actions of $V$

This continues my question about prefix-continuous bijections (since the answer was "yes").
Notation and conventions: Let $A$ be a finite alphabet and $L \subset A^*$ a language. Let $G$ be a group. ...

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**1**answer

128 views

### Is there a prefix-continuous bijection between finite words and eventually zero words?

Let
$$ X = \{x \in \{0,1\}^{\omega} \;|\; \exists m: \forall i \geq m: x_i = 0\} $$
(one-way infinite eventually zero words). Let $\{0,1\}^*$ denote the finite (not necessarily nonempty) words over $\{...

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179 views

### hyperbolic quotient of hyperbolic group

I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...

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68 views

### Large gaps in the norm of a subgroup and its centraliser

Take an infinite finitely generated group $G$ with an infinite subgroup $N$ which has an infinite centraliser $Z = Z_G(N)$.
Let $S$ be some [symmetric] generating set of $G$ and for $g \in G$,
...

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157 views

### Image of the mapping class group of surfaces into automorphism group?

Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...

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62 views

### QI-closure of $\mathrm{NA}\times\mathrm{NA}$

Suppose we know the following about a class of groups $\mathcal{G}$.
If $G$, $H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$.
If $G \in \mathcal{G}$, $G$ is f.p., and $G$ is ...

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127 views

### Reference request: Name or use of this group of diffeomorphisms of the disc

Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...

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145 views

### Convergence of Fuchsian groups and existence of suitable homeomorphisms

Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...

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118 views

### Membership problem in general linear group

This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am.
Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...

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65 views

### Finitely generated uniformly amenable groups

Keller in "Amenable groups and varieties of groups" introduces uniformly amenable groups as groups such that
there is a function $a: ]0,1[ \times \mathbb{N} \to \mathbb{N}$ such that
for any finite ...

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123 views

### For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?

Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$,...

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**1**answer

193 views

### Powers of the Euler class, torsion free subgroup of Homeo($S^1$)

For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...

**15**

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265 views

### Can a torsion-free group be quasi-isometric to a torsion group?

I have looked around in the literature on group theory and geometric group theory and this looks to be an open question as far as I can tell (by torsion group, I mean as usual a group in which every ...

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**1**answer

89 views

### Is a cosparse action on a CAT(0) cube complex an essential action?

Let $X$ be a CAT(0) cube complex.
(From Sageev and Wise's Cores for Quasiconvex actions)
A group $G$ acts cosparsely on a CAT(0)-cube complex $X$ if there exists a compact space $K$ and finitely many ...

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64 views

### The divergence function and quasigeodesics in $\delta$-hyperbolic spaces

(From Bridson and Haefliger's Metric Spaces of Non-positive Curvature) Let $X$ be a metric space. A map $e: \mathbb{N} \rightarrow \mathbb{R}$ is said to be a divergence function for $X$ if the ...

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121 views

### When existence of loxodromic, WPD elements implies an action is acylindrical

Definitions
Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries.
Recall that an action of $G$ on $X$ is called acylindrical if the ...

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252 views

### Existence of a quasi-isometric residually finite group?

It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...

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147 views

### The stabilizers of the canonical boundary action of hyperbolic groups

My question is that
Is every stabilizer of the canonical boundary action of a hyperbolic group on its Gromov boundary a finitely generated group?
I guess every stabilizer is a (finitely generated) ...

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**1**answer

93 views

### Bounding the size of the conjugating elements given the Dehn function

I am learning a little bit about Dehn functions of group presentations and I came across a question that is probably pretty basic but that I was giving me trouble. I'll set some notation but ...

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50 views

### A Jacobian of the Cayley graph

Let $\pi$ be a finitely generated discrete group. Let $F$ be a finitely generated free group with an epimorphism to $\pi$. Let $K$ be the kernel of the epimorphism. Consider the Abelianization of $K$. ...

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74 views

### Quasi-isometries and E-unitary inverse semigroups

Let $S = \langle K\rangle$ be a finitely generated inverse semigroup, where $K \subset S$ is a fixed, finite and symmetric set of generators.
Preliminaries: Recall that we say that $s, t \in S$ are $\...

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36 views

### Examples of groups admitting a proper $1$-cocyle for a bounded representation

A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...

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119 views

### Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...

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261 views

### On what proper Gromov hyperbolic space does a free product act?

Per Bowditch, a group is relatively hyperbolic if it acts geometrically finitely on a proper geodesic Gromov hyperbolic space.
A free product of two (or finitely many) finitely generated groups is ...

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221 views

### Thickness and hierarchical hyperbolicity

Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here.
I've heard that it is open ...

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50 views

### Quotient of Euclidean space with maximal volume growth

Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...

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147 views

### Property (T) for pairs

I was reading the excellent book by Bekka, de la Harpe, and Valette (link at Bekka's page), and on its list of open questions I was looking at p300 Question 7.7:
$L\subset K\subset H\subset G$.$H,K,...

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348 views

### Constant Martin kernel and amenability

Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...

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227 views

### Commutator algorithm

Let $M \in \mathrm{SL}(2, \mathbb{Z}).$ Is there an efficient algorithm to write $M$ as a commutator (group commutator, not algebra commutator) [or fail if this is impossible]?
Addendum: answering ...

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165 views

### Dualizing module for $\operatorname{Aut}(F_n)$

In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...

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187 views

### Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space

Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂𝑋 be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich https://...

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231 views

### Finite quotients of surface braid groups

Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$.
There is a braid $A_{12} \in \Sigma_b$ ...

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210 views

### Status of questions in “Group Actions on $\mathbb{R}$-trees”?

Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...

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263 views

### Is a product of Følner sets Følner?

Let $G$ be an amenable (countable, discrete) group and let $F_1,F_2,...,F_n,...$ and $G_1,G_2,...,G_n,...$ be two Følner sequences. Is the product sequence (i.e. the sequence $(H_n)$ where $H_n$ is ...

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171 views

### Universal cover or Bass-Serre tree: difference between definitions given by Bass and Serre

Let $(\mathbb G,\Gamma)$ be a graph of groups.
A $G$-path from $u_0$ to $u$ is
$$g_0e_1g_1\cdots e_{n}g_n,$$
where $e_1\cdots e_{n}$ is a walk in $\Gamma$ from $u_0$ to $u$ and each $g_i\in G_{s(e_i)}$...

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200 views

### Examples of non-cubulated hyperbolic groups

What is known regarding which hyperbolic groups are cubulated?
I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.
My impression is that not all ...

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78 views

### Intuition behind the local limit theorem in hyperbolic groups

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Denote by $X_n$ the induced random walk. Finally, let $p_n=\mu^{*n}(e)=P_e(X_n=e)$. The local limit ...

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229 views

### Growth rates of surface groups

I'm looking for readable references on calculating the growth rates of surface groups.
There's some approach done briefly in page 159 of de la Harpe's "Topics in Geometric Group Theory", who cites:
...

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283 views

### Is property FA of Serre known for $SL_2(\mathbb{Z}[i])$ and $SL_2(\mathbb{Z}[\zeta_3])$

In Serre's book Trees [Se, p. 68] it says:
3) For $SL_2$ the situation is different. It is clear that
$SL_2(\mathbf Z )$ does not have property (FA). It is the same with
$SL_2(A)$ when $A$ is ...

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73 views

### Abelian-by-cyclic subgroups of exponential growth solvable groups

I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...

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**1**answer

151 views

### Asymmetry of outer space - injectivity radius

I'd like to ask a question on "Asymmetry of Outer Space" by Yael Algom-Kfir & Mladen Bestvina.
In Example $2$, page $4$ it says "Note that in this case the asymmetry can be explained by the fact ...

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**1**answer

336 views

### Does Thompson's group F have Yu's property A?

As far as I know, it is unknown whether Thompson's group F has Yu's Property A (that is, whether it is exact) or not. See for instance this MO question. The question is said to be open at several ...

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203 views

### Multiplication in Thompson's Group F

Does there exist an algorithmic way to multiply two elements of the Thompson Group F together? Specifically when looking at it from the perspective of pairs of binary trees. To multiply two elements ...

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416 views

### virtual isomorphism of groups is an equivalence relation

I am reading the book: Geometric Group Theory by
Cornelia Druţu and Michael Kapovich With an Appendix by Bogdan Nica. https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf. On page 125 we have:
*Def: We ...

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99 views

### Examples of non-uniform lattices in products of trees

Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete.
So as a vague general question, what ...

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168 views

### Subgroups of torsion-free hyperbolic groups versus subgroups of hyperbolic groups

Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such ...

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151 views

### Equivalence of harmonic measures on hyperbolic groups

Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, ...

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246 views

### What does the free action of a surface group on an R-tree look like?

Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, ...

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158 views

### QI but not ME : not finitely presented groups!

I would like to know the examples of two groups which are not finitely presented and are quasi-isometric (QI), but they are not measured equivalent (ME) (in the sense of Gromov).
In the literature, ...

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**1**answer

341 views

### Characterization of countable groups as groups with a left-invariant distance with finite balls

In their book "Metric geometry of locally compact groups", Yves Cornulier and Pierre de la Harpe gave, the following characterization of countable groups: $G$ is countable if and only if it has a ...