# 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

**6**

votes

**0**answers

76 views

### Integral of Gromov product

Consider a geodesic, hyperbolic metric space $(X,d)$. For $x,y \in X$, consider $D(x,y)$ the infimum, over all $1$-Lipschitz paths $p:[a,b] \rightarrow X$ from $x$ to $y$, of the integral $\int_a^b e^{...

**6**

votes

**0**answers

140 views

### Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...

**7**

votes

**1**answer

107 views

### Going up of an amalgamated decomposition of a subgroup of finite index

Let $G$ be a finitely presented group and H a subgroup of index $n$ in $G$. Suppose that H has a non-trivial decomposition as amalgamated product, say $H = A \ast_U B$. I am wondering about the ...

**4**

votes

**2**answers

119 views

### Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$

I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty:
Specifically, ...

**1**

vote

**0**answers

47 views

### Relations between Omega, Local Indicable and Right Orderable groups

We know that the set of Right-Orderable groups $RO$, is contained in the set of $\Omega$- groups (Read it from "A Note on Group Rings of Certain Torsion-Free Groups" Burns-Hale).
A Group $G$ is a ...

**-1**

votes

**1**answer

73 views

### Nilpotent subgroups of the direct limit of $GL_n(\mathbb{Z})$ with arbitrarily large finite subgroup

We embeds $GL_n(\mathbb{Z})$ in $GL_{n+1}(\mathbb{Z})$ by identifying $A \in GL_n(\mathbb{Z})$ with $\begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix} \in GL_{n+1}(\mathbb{Z})$. Let $GL_\infty(\...

**4**

votes

**0**answers

104 views

### Subgroup membership problem for Noetherian groups

I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group,
\begin{...

**3**

votes

**0**answers

73 views

### Length functions on Teichmuller space with constant difference

Let $S$ be a closed oriented surface of genus $g\geq 2$. Let $\mathcal{T}$ be the corresponding Teichmuller space. Given a free homotopy class of closed curve $[\gamma]$ we can define the length ...

**9**

votes

**0**answers

89 views

### Can every non-inner automorphism of a group with residually finite outer automorphisms be realised as an non-inner automorphism of a finite quotient?

First some motivation: most proofs that show that the group of outer automorphisms is residually finite do not only show that the subgroup of inner automorphisms is closed in the profinite topology, ...

**4**

votes

**1**answer

129 views

### Infinite finitely presented simple group (or more generally with trivial profinite completion) that is not amalgamated free product

As is described in the title. Is there a finitely presented group $G$, with trivial profinite completion $\widehat{G}=0$, which is not amalgamated free product?
For example, the famous example Higman ...

**6**

votes

**0**answers

65 views

### Stable commutator lengths of pseudo-Anosovs

Does anyone have an example of a pseudo-Anosov mapping class for which the stable commutator length is known exactly?

**4**

votes

**0**answers

124 views

### Non-algebraic quasi-isometric embeddings

What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...

**10**

votes

**1**answer

293 views

### Residually finite group surjective to nonresidually finite group with finitely generated kernel

As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: G\rightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $...

**5**

votes

**0**answers

103 views

### Automorphism groups of cocompact Fuchsian groups as mapping class groups

Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...

**4**

votes

**0**answers

146 views

### 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, ...

**3**

votes

**1**answer

129 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)=...

**5**

votes

**0**answers

127 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 $|g|$ ...

**6**

votes

**4**answers

511 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 ...

**2**

votes

**1**answer

251 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|...

**5**

votes

**2**answers

173 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 ...

**9**

votes

**0**answers

241 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 $...

**12**

votes

**2**answers

523 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 ...

**1**

vote

**0**answers

114 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 $...

**3**

votes

**1**answer

106 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?

**1**

vote

**1**answer

166 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 $X$?

**8**

votes

**1**answer

175 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 ...

**31**

votes

**1**answer

938 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 ...

**7**

votes

**4**answers

262 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 $G$ ...

**5**

votes

**1**answer

227 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 ...

**26**

votes

**1**answer

431 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 ...

**10**

votes

**0**answers

266 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 ...

**11**

votes

**1**answer

250 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 ...

**5**

votes

**0**answers

47 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 ...

**9**

votes

**1**answer

323 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"...

**2**

votes

**1**answer

184 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 ...

**8**

votes

**0**answers

122 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 ...

**19**

votes

**1**answer

421 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 ...

**5**

votes

**1**answer

202 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 ...

**5**

votes

**0**answers

125 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-...

**4**

votes

**2**answers

310 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?

**5**

votes

**1**answer

287 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 $B=\bigsqcup_{i=1}^na_iA_i$.
...

**3**

votes

**1**answer

179 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 ...

**10**

votes

**1**answer

138 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 ...

**3**

votes

**1**answer

145 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 ...

**5**

votes

**2**answers

216 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 ...

**11**

votes

**0**answers

198 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$^*$...

**6**

votes

**0**answers

205 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 $\...

**4**

votes

**2**answers

140 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 ...

**16**

votes

**0**answers

251 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 ...

**2**

votes

**1**answer

110 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 $...