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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### Uniqueness of the boundary of a hierarchically hyperbolic group

Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...

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50 views

### Which $CAT(0)$-polygonal complexes are median spaces?

$CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag.
Which $CAT(0)$-polygonal complexes with appropriate ...

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54 views

### Do $CAT(0)$-polygonal complexes have hyperplanes?

$CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag.
Do $CAT(0)$-polygonal complexes have "hyperplanes" ...

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**1**answer

116 views

### Are $CAT(0)$-polygonal complexes median spaces?

A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)=...

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122 views

### The preimage of bounded real intervals under homomorphisms on hyperbolic groups

Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ ...

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480 views

### What is a geodesic in Outer space?

The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different ...

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**1**answer

236 views

### About the growth rate of a group

Let $G$ be a f.g. group and $d$ be a word metric w.r.t. a symmetric generating set. For $g\in G$, define $|g|:=d(g,e)$, where $e$ is the group identity. For $k\in\mathbb N$, put
$$n_k:=\#\{g\in G: |g|...

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167 views

### Codimension-1 subgroups of 3-manifold groups

Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$.
Do all ...

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235 views

### Sharpness of the $1/6$-constant in the Cancellation Theorem

I originally posted this question over at Stackexchange, before realising it is much better suited for Overflow:
Let $\langle \; S \; | \; R \; \rangle$ be a presentation of a group $G$ with a set $...

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500 views

### Modern references on hyperbolic groups

Several good references dedicated to hyperbolic groups have been written during the 90s, including:
Hyperbolic groups, written by M. Gromov.
Géométrie et théorie des groupes : les groupes ...

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111 views

### Codimension-1 subgroups and submanifolds

Let $G$ be a finitely generated group and let $H$ be a subgroup of
$G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$.
Let $...

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**1**answer

101 views

### Group acting on a CAT(0) cube complex then acting also on a tree

If a group $G$ acts on a CAT(0) cube complex, then does $G$ act on a simplicial tree?

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163 views

### Group action on quasi-isometric geodesic metric space [closed]

If a group $G$ acts on a geodesic metric space $X$, then does $G$ act on a geodesic metric space $Y$ which is quasi-isometric to $X$?

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172 views

### If a group $G$ has decidable word problem, must it have a decidable square problem?

My question is a refinement of this one about 'efficient' construction of square elements: If the word problem for a (finitely generated, finitely presented) group is decidable, must the 'square ...

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114 views

### Representing curves using words

I am trying to understand how in this paper https://arxiv.org/abs/1412.0101 he represents curves with words. This is on page 10 of the paper.
Assume that two piecewise smooth closed curves $\gamma_1$ ...

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**1**answer

924 views

### Is this conjecture strictly weaker than P=NP?

My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...

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**4**answers

259 views

### Lattices of PU(n,1) with large abelianization

I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...

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**1**answer

221 views

### Cayley graph properties

Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected ...

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**1**answer

425 views

### What is the minimal dimension of a complex realising a group representation?

This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).
Many interesting integral ...

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262 views

### Amalgamated product of automatic groups

In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...

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**1**answer

246 views

### Genus of Cayley graph of $A_5$ with two generators of order 5

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. I am wondering what is its graph genus (orientable or non-orientable). The best I could get by trial and error is an ...

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45 views

### Computing centralizers of finite sets in right angled Artin groups (RAAGs) / partially commutative groups / graph groups

This question concerns right angled Artin groups (RAAGs), also called partially commutative groups or graph groups.
A student of mine, Adi Ben-Zvi, needs for an algorithm in RAAGs, a subalgorithm ...

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**1**answer

322 views

### Divergence of Groups and Metric Spaces

Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...

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**1**answer

171 views

### Quotient groups of the lower central series of a surface group

In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
Q. What are the quotient ...

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118 views

### Geodesics between boundary points of a hyperbolic space

Let $X$ be a (not necessarily proper) hyperbolic space. Following Gromov, we define the boundary of $X$ as the set of equivalence classes of sequences convergent at infinity. In general, it is not ...

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**1**answer

413 views

### Can a hyperbolic, one ended, one relator group, have a shorter trivial word?

Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the ...

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**1**answer

198 views

### Conformal boundary and cusp of figure-8 complement

As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...

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122 views

### Filling points to a simplex in models for EG

I have a question which is related to higher Dehn functions of groups.
I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...

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307 views

### Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup?

Are there any known examples of groups that virtually split that don't have a codimension-1 subgroup?

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**1**answer

284 views

### Can infiniteness of finitely generated groups be read by a “paradoxical” decomposition?

(Edit) Let $G$ be a group. Two subsets $A,B$ of $G$ are said to be equidecomposable if there exists a finite partition $A=\bigsqcup_{i=1}^nA_i$ and $a_i\in G$ such that $B=\bigsqcup_{i=1}^na_iA_i$.
...

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**1**answer

176 views

### Classifying/characterising groups acting on CAT(0)-cube complexes

What are some potential approaches to classifying/characterising (finitely presented?) groups that act (without fixed points and potentially more conditions) on CAT(0)-cube complexes? As so many ...

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137 views

### Minimal length presentations of cyclic groups

By the length of a finite presentation I mean the sum of the lengths of the relators. I am interested in knowing what the minimal length of a presentation of $\mathbb{Z}/n\mathbb{Z}$. I'm even more ...

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**1**answer

140 views

### Normal generating set for the intersection of two normal subgroups of a surface group

Let $G = \left< a_1,b_1, ... , a_g, b_g | [a_1,b_1] \cdots [a_g,b_g] \right>$ be the fundamental group of a surface of genus $g$. Let $N_1$ and $N_2$ be two normal subgroups of $G$ that are ...

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202 views

### generators for the handlebody group of genus two

Is the handlebody group of genus two surface generated by Dehn twists along properly embedded disks and annuli?
Are there alternative ways to describe a set of generators that are conceptually simple ...

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193 views

### Does Thompson's group $V$ have property AP?

Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...

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195 views

### Generators for commutator subgroup of surface group

Let $\pi_1(\Sigma_g) = \langle\text{$x_1,\ldots,x_{2g}$ $|$ $[x_1,x_2]\cdots[x_{2g-1},x_{2g}]$}\rangle$ be a surface group. Can anyone tell me an explicit free basis for the commutator subgroup of $\...

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136 views

### Nielsen-Thurston decomposition from the product of Dehn twists

Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is
Given an element as a product of Dehn twist, is it possible ...

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244 views

### Finitely generated groups with Hölder-exotic space of ends?

The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...

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**1**answer

109 views

### Teichmuller uniqueness theorem with marked points

Let $S$ be a genus $g$, $g > 1$ Riemann surface, and let $h \colon S \to S$ be a homeomorphism of $S$. We denote by $[h] \in \text{Map}(S)$ the corresponding element of the mapping class group of $...

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**1**answer

318 views

### Parabolic subgroups of relatively hyperbolic and CAT(0) groups

Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...

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**1**answer

231 views

### Relative/acylindrical hyperbolicity of free-by-cyclic groups

Is this statement true?
Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.
If $\varphi$ ...

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**1**answer

180 views

### Do commutator functor and intersection commute?

For two subgroups $A, B$ in $G$,
$[A,A] \cap [B,B] = [A\cap B, A \cap B]$?
At least, if $G$ is free, is the left contained in the right?

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169 views

### Elements of a group than can achieve all orders in its quotients

Let $G$ be a group. For different purposes I am now interested in a quite natural property that the elements in $G$ may or may not have, and I would like to ask if there is a standard terminology for ...

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**3**answers

237 views

### Graphs of groups with homomorphisms not necessarily injective

I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general ...

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111 views

### 2-generator subgroups of an Artin group of small type

Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...

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332 views

### Random walk uniformly hitting a compact set

Suppose $G$ is a locally compact compactly generated group. Let $\mu$ be a probability measure that is:
Adapted to $G$, i.e. there is no proper subgroup $H$ such that $\mu(H)=1$.
Symmetric, i.e. $\...

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484 views

### Subgroup of hyperbolic group generated by non-torsion elements

Let $G$ be a hyperbolic group. I know that it is an open problem whether $G$ has a torsion-free subgroup of finite index. But if we let $N$ be the subgroup of $G$ generated by its non-torsion elements,...

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150 views

### Divergence of Green function of random walks at spectral radius

Let $P=(p(x,y))_{x, y\in N}$ be the transition matrix over countable states $N$.
Consider the generating Green function $G(x, y|t)=\sum_{0}^{\infty} p^n(x, y) t^n$, where $p^n(x,y)$ is the $(x,y)$-...

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**1**answer

77 views

### Harmonicity of the Martin kernels

Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Consider the Green function $G(x,y)=\sum_{n\geq 0}\mu^{*n}(x^{-1}y)$, where $\mu^{*n}$ is the $n$th ...

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175 views

### Rational stable translation length

Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...