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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14
votes
3answers
503 views

Failure of Mostow rigidity in dim. 2

I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question: (1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...
10
votes
1answer
188 views

Cantor-Bernstein for quasi-isometric embeddings?

Suppose that two finitely generated groups quasi-isometrically embed into each other. Does it follow that the two groups are quasi-isometric? Recall that a quasi-isometry is a quasi-isometric ...
5
votes
2answers
258 views

Which groups are LERF?

A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
2
votes
0answers
140 views

Marshall Hall's theorem for surface groups [closed]

Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$ Let $H \leq \Gamma_g$ ...
4
votes
0answers
86 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 ...
6
votes
1answer
143 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 ...
3
votes
1answer
161 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 ...
10
votes
1answer
375 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
0answers
113 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
0answers
55 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
1answer
122 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 ...
13
votes
2answers
346 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 = ...
4
votes
2answers
210 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 ...
5
votes
1answer
189 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
0answers
142 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 ...
4
votes
0answers
80 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 ...
6
votes
1answer
251 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
1answer
295 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
0answers
138 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
1answer
195 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 ...
1
vote
0answers
139 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 ...
2
votes
0answers
159 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 ...
4
votes
1answer
108 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
1answer
237 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 ...
6
votes
0answers
131 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?
2
votes
0answers
207 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 ...
7
votes
1answer
299 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
1answer
233 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 ...
-2
votes
1answer
95 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
0answers
82 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 ...
6
votes
1answer
174 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
1answer
139 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
1answer
196 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
3answers
319 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
1answer
653 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
0answers
160 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
1answer
217 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 $H$ ...
6
votes
0answers
106 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
1answer
301 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 $ ...
1
vote
1answer
111 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
1answer
127 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 ...
3
votes
0answers
206 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 ...
3
votes
0answers
67 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
3answers
191 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
0answers
106 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
1answer
140 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
3answers
599 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
0answers
109 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, ...
11
votes
2answers
328 views

Algorithms in hyperbolic groups

I'm stuck in some algorithms in hyperbolic groups, which may be rather simple. Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can ...
3
votes
0answers
95 views

Does there exists a finitely presented group with Dehn function > n^3 and all asymptotic cones simply connected

It is well known that all asymptotic cones simply connected implies polynomial Dehn function (Gromov). It is also well known that quadratic Dehn function implies all asymptotic cones simply connected ...