# 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

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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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, ...
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 ...