**2**

votes

**0**answers

58 views

### Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...

**7**

votes

**2**answers

506 views

### Translation length functions of non-simplicial trees

Let $G$ be a finitely generated group. By a theorem of Culler and Morgan, the set of non-abelian (not necessarily simplicial) minimal $\mathbb{R}$-trees with isometric $G$-action injects into the ...

**3**

votes

**1**answer

134 views

### δ-hyperbolic space

It is known that A δ-hyperbolic space is a geodesic metric space in which every geodesic triangle is δ-thin. (http://en.wikipedia.org/wiki/%CE%94-hyperbolic_space)
The question is that if we remove ...

**6**

votes

**1**answer

128 views

### Is the following module over a group ring necessarily infinitely generated?

Suppose $\Gamma$ is a (finitely presented, but this is probably irrelevant) group, and $M$ is a finitely generated (EDIT: finitely presented) module over $\mathbb{Q}\Gamma$ which is ...

**4**

votes

**0**answers

163 views

### Computing the pro-solvable closure of a finitely generated subgroup of a free group

The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...

**9**

votes

**1**answer

329 views

### Holomorphic cusp forms and cohomology of GL(2,Z)

Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...

**6**

votes

**0**answers

79 views

### CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...

**0**

votes

**0**answers

52 views

### Bases of surface groups with length restrictions

This question asks for a generalization of Bases of surface groups following the notation and definitions given therein.
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, $B$ a surface basis of ...

**4**

votes

**1**answer

111 views

### Ends of quotients of Coxeter Groups

Let C(p,q) be the Coxeter group:
$C(p,q):= \langle a,b,c\hspace{1mm}|\hspace{1mm} a^2,b^2,c^2,(ac)^2,(ab)^p, (bc)^q \rangle$
for integers $p,q$ s.t. $\frac{1}{p}+\frac{1}{q}<\frac{1}{2}$. This ...

**12**

votes

**1**answer

301 views

### Asymmetric metrics and cohomology

If $(X,d)$ is a metric space and $f : X \rightarrow \mathbb{R}$ is a Lipschitz function with Lipschitz constant $k < 1$, then the function
$$
D(x,y) := d(x,y) + f(y) - f(x)
$$
defines an asymmetric ...

**4**

votes

**2**answers

188 views

### A Karrass-Solitar theorem for surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...

**13**

votes

**2**answers

303 views

### Minimal number of generators for $GL(n,\mathbb{Z})$

$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = ...

**5**

votes

**1**answer

174 views

### Bases of surface groups

Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...

**1**

vote

**0**answers

126 views

### Accessible subgroups of free groups

Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...

**3**

votes

**0**answers

66 views

### Nice minimal embeddings of large finite groups into compact Riemannian manifolds

The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...

**7**

votes

**2**answers

358 views

### Kaplansky's idempotent conjecture for Thompson's group F

Let $K$ be a field and $G$ be a torsion-free group. Kaplansky's idempotent conjecture states that the group ring $K[G]$ does not contain any non-trivial idempotent, i.e. if $x^2=x$ then $x=0$ or ...

**6**

votes

**1**answer

242 views

### Amenability as a geometric property

Let $G$ be a discrete, finitely generated, and amenable group. Let $H$ be a group which is quasi- isometric to $G$. Is $H$ amenable?

**11**

votes

**1**answer

277 views

### What are smallest finite images of triangle groups?

Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, ...

**3**

votes

**0**answers

126 views

### $S^{3}$-valued harmonic analysis

Edit:
Note that $S^{3}$ with the quaternion operation is a group. For a locally compact Abelian group $\Gamma$ we consider
$$\tilde{\tilde{\Gamma}}=\{\phi:\Gamma \to S^{3} \mid ...

**10**

votes

**1**answer

160 views

### Torsion-free group that is not of type F but is virtually of type F

Recall that a group $G$ is of type F if there exists a compact $K(G,1)$.
There are many examples of groups which are not of type F but which are virtually of type F, that is, they have finite-index ...

**4**

votes

**1**answer

97 views

### Stabiliser of the lamination of a free group - Invariant subgraphs

I am studying the paper "Laminations, trees, and irreducible automorphisms of free groups" of Bestvina, Feighn and Handel. But I found a note in the paper "Stabilisers of $\mathbb{R}$-trees with ...

**10**

votes

**1**answer

197 views

### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...

**1**

vote

**0**answers

120 views

### Reference on calculation of 2nd cohomology group

Let $G$ be a finitely generated, infinite, countable discrete nonamenable group with zero first Betti number, I.e., $H^1(G, \ell^2(G))=0$, e.g., $G=F_2\times F_2$, the product of free groups of two ...

**1**

vote

**0**answers

189 views

### Kaplansky Idempotent conjecture and Extension theory

We consider the Idempotent Kaplansky conjecture with $\mathbb{C}$- coefficients, that is the problem of nontrivial idempotents for group algebra $\mathbb{C}\Gamma$ where $\Gamma$ is a torsion free ...

**2**

votes

**0**answers

104 views

### Finitely generated groups non-embeddable into $L_1(0,1)$

I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group ...

**15**

votes

**9**answers

3k views

### The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$)

Is there someone who can give me some hints/references to the proof of this fact?

**6**

votes

**0**answers

117 views

### Homology groups of Noetherian groups

Let $G$ be a Noetherian group. Is $H_n(G,\mathbb{Z})$ finitely generated? Do we know the above for the special cases $n=2,3$ even?

**7**

votes

**1**answer

261 views

### Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

Suppose $G$ is a discrete group given by finitely many generators with finitely many relations. Can the homology groups $H_i(G, \mathbb{Q})$, or equivalently $H_i(BG, \mathbb{Q})$ (topological ...

**2**

votes

**1**answer

230 views

### Can a closure make the index finite?

Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of ...

**-1**

votes

**1**answer

84 views

### Is there some lattice not rigid

I heard that in complex hyperbolic space setting for example CH2, there is some deformation of lattice nontrivial. What confused me is it seems contradicting Mostow Rigidity. Could someone explain ...

**0**

votes

**0**answers

78 views

### Bases and transversals

Let $F$ be a free finitely generated group, $L \leq H \leq F$ subgroups of finite index.
Given bases $B$ of $F$ and $C$ of $H$, must there be a Schreier transversal $T$ for $H$ in $F$ such that the ...

**5**

votes

**1**answer

159 views

### Orbit spaces of crystallographic groups

In their paper "On Three-Dimensional Space Groups", Conway et al. write
Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups ...

**1**

vote

**1**answer

128 views

### Products of subgroups that generate a finite group

Consider the following general problem. There is a finite group $G$ and $H_1,H_2 < G$. Suppose we know that $\langle H_1, H_2 \rangle = G$, i.e. $G$ is generated by $H_1$ and $H_2$. Denote by $n_0$ ...

**2**

votes

**1**answer

185 views

### Making a profinite group free

Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group?
For abstract groups the ...

**3**

votes

**3**answers

269 views

### Stallings' Theorem for free products of groups

There is a well known theorem which states that:
Theorem(Stallings):
For any immersion $f$ from a finite graph $D$ to $G$ there is a finite-sheeted covering space $D '$ of $G$ that extends $f$. ...

**4**

votes

**1**answer

609 views

### solvable word problem without algorithm

Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...

**6**

votes

**0**answers

148 views

### “Twisted” Lyndon equation in a free group

In 1959, Lyndon showed that in a free group, the equation $u^2v^2=w^2$ has only commuting solutions: $uv=vu=w$. Is there in the litterature any information about the following "twisted" version of the ...

**3**

votes

**1**answer

190 views

### Bases of free groups

Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...

**2**

votes

**2**answers

166 views

### Commutator width of soluble Baumslag Solitar groups

Do the soluble Baumslag-Solitar groups have finite commutator width? A soluble Baumslag-Solitar group is given by a presentation of the from
BS(1,m) = $<a,b \mbox{ }| \mbox{ } a^{-1}ba = ...

**6**

votes

**0**answers

99 views

### Two-relator products of cyclic groups

In "A proof of the Scott–Wiegold conjecture on free products of cyclic groups" Howie proved that every one-relator product of three cyclic groups is nontrivial. Is there a now proven theorem that says ...

**1**

vote

**1**answer

287 views

### Kropholler's Conjecture and 3-manifolds

Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial H-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that $ ...

**3**

votes

**0**answers

177 views

### Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...

**1**

vote

**1**answer

110 views

### the lower bounded of metrics on a group

Let $G$ be a finitely generated group, S is a set of generators. If $\forall s\in S, n\in \mathbb{Z}$, $\exists C>0$ such that $|s^n|_S\ge C|n|$, does it imply $\forall$ infinite order element ...

**4**

votes

**1**answer

112 views

### Action of the isometry group of the hyperbolic 5-space

We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+_{5,1}(\mathbb{R})/SO_5(\mathbb{R})=SL_2(\mathbb{H})/Sp^*_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the ...

**1**

vote

**0**answers

52 views

### L_2-Betti numbers of random groups

Are there any results in calculating $l_2$-Betti numbers of random groups in the Gromov density model?
I can deduce myself a few facts:
If a group has Property (T) then the first $l_2$-Betti ...

**2**

votes

**2**answers

120 views

### Combination of Stallings theorem and Dunwoody's accessibility

I am interested in applications of the combination of Stallings theorem and Dunwoody's accessibility, which can be summarized as follow:
Theorem: Let $G$ be a finitely presented group. Then $G$ is ...

**4**

votes

**0**answers

101 views

### Regularity of polynomial growth of groups

Let $G$ be a finitely generated group of polynomial growth. This means that the size $B_n$ of the ball of radius $n$ satsifies:
$$
A n^d \leq B_n \leq Bn^d
$$
for some constants $A$, $B$.
My question ...

**4**

votes

**1**answer

111 views

### Rationality of translation lengths in hyperbolic groups

Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...

**6**

votes

**3**answers

555 views

### Links between Geometric Group Theory and Number Theory

Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.

**3**

votes

**0**answers

99 views

### Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...