**5**

votes

**1**answer

213 views

### Cocycles for right- and left- regular representations on $\ell_2(G)$

Using the standard notation, let $\lambda_gf(x)=f(g^{-1}x)$ be the left regular representation and $\rho_gf(x)=f(xg)$ be the right regular one, acting on the space $V$ of complex-values functions on ...

**3**

votes

**1**answer

209 views

### Flows in word-hyperbolic groups

I was wondering if there is a good notion of flows in word-hyperbolic groups (maybe I should say flows in the Cayley graphs of word-hyperbolic groups).
More precisely, I wonder if there is an ...

**10**

votes

**2**answers

379 views

### Invariant free factor of a free group

Let $F_n=F\ast F'$ be a free splitting of the free group $F_n$ and $\phi\in Aut(F_n)$. The free factor $F$ is said to be invariant under $\phi$ if $\phi(F)\subseteq F$.
I recently wondered if this ...

**4**

votes

**2**answers

258 views

### Primitive subwords in a free group of rank 2

I am not sure yet about what I exactly need to prove, but I guess I can formulate a rough statement similar to the following:
Suppose $w\in F_2$ is a primitive word whose length is big enough. Then ...

**4**

votes

**3**answers

312 views

### General properties of free-by-cyclic groups

I admit this is a very broad question, but I am looking for general properties of [finitely generated free]-by-[infinite cyclic] groups. More precisely, what are some properties that the groups ...

**4**

votes

**2**answers

303 views

### convergence action on the boundary of hyperbolic groups

Let G be a word-hyperbolic group acting on its boundary, which is homeomorphic to $S^n$ (n-sphere), effectively. Does this imply that G acts on the boundary as a convergence group of $S^n$?
If this ...

**2**

votes

**0**answers

109 views

### Local curvature in a Cayley complex

I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local ...

**11**

votes

**6**answers

1k views

### Understanding groups that are not linear.

I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:
What are some interesting ...

**5**

votes

**1**answer

341 views

### Does $C'\left(\frac{5}{11}\right)$ imply exponential growth?

I came across this rather week small cancellation condition $C'\left(\frac{5}{11}\right)$ of a group $G$. It has been proved that $C'\left(\frac16\right)$ is enough for $G$ to contain free subgroups. ...

**6**

votes

**2**answers

255 views

### Dehn function for undistorted subgroups of a product of free groups

Let $G$ be a finitely generated subgroup of a product of two finite rank free groups $F_m \times F_n$. If there is a Lipschitz retraction $F_m \times F_n \to G$ with respect to word metrics, then $G$ ...

**3**

votes

**1**answer

207 views

### Lowering metrics to finite index subgroups

Let $G$ be an infinite, countable, finitely generated group. Let $H$ be a finite index subgroup of $G$. Let $S$ be a finite, symmetric set of generators of $G$, and let $d(\cdot,\cdot)$ be the word ...

**1**

vote

**2**answers

370 views

### Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that ...

**5**

votes

**0**answers

207 views

### Any method to detect subgroup generated by a subset of the generators from its presentation

I have met the following problem. A group $G$ is given as follows
$G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$
Is the subgroup generated by $y$ and $t$ just the ...

**2**

votes

**1**answer

584 views

### Fundamental Lemma Of Geometric Group Theory

I'll be delighted to get some help in understanding the proof of the first theorem here:
http://www.math.utah.edu/~malone/QI/notes.pdf
"If G acts geometrically on X and Y (proper geodesic metric ...

**3**

votes

**1**answer

644 views

### French resources for (Geometric) Group Theory

I am looking for ways to improve my mathematical French while learning more material about either finite group theory or geometric group theory. In particular, I would love to find a French equivalent ...

**20**

votes

**2**answers

863 views

### Asymptotics of the growth rate of a group

Let $\Gamma$ be a finitely generated group of exponential growth and $gr(S)=\lim_{k\rightarrow \infty} \sqrt[k]{|B_k(S)|}$ be the growth rate of $\Gamma$ with respect to the generating set $S$. I am ...

**1**

vote

**2**answers

174 views

### Conjugated elements in amalgameted Product

Hello!
Let $G=A\underset{C}\star B$ be an amalgamated Product.
Let $a\in A$. If a is conjugated to an Element $b\in B$, then $a$ is conjugated to an Element $c\in C$. The Question is: Why is that ...

**11**

votes

**2**answers

502 views

### Does every group grow either polynomially or superpolynomially?

I am reading an introduction to growth of groups. The notions of polynomial and superpolynomial growth are introduced, as are exponential and subexponential growth.
I can prove that the growth of a ...

**3**

votes

**2**answers

389 views

### Free groups as quotients of hyperbolic groups

Given any infinite non-elementary hyperbolic group $G$, a theorem of Gromov asserts that there is a subgroup of $G$ isomorphic to a non-abelian free group on two generators.
Is there a similar result ...

**9**

votes

**3**answers

485 views

### Finite subgroups of relatively hyperbolic groups

It is well known that in a given $\delta$-hyperbolic group there are only finitely many conjugacy classes of finite subgroups. This is clearly false for relatively hyperbolic groups since we have no ...

**7**

votes

**1**answer

212 views

### Asymptotics of the number of required Dehn relators in hyperbolic groups

If $G = \langle X | R \rangle$ is a $\delta$-hyperbolic group presentation, then Dehn's algorithm provides a linear time solution to the word problem, but the linear constant is horribly exponential ...

**16**

votes

**1**answer

729 views

### Amenable groups of deficiency $1$

Let $G=\langle X;R\rangle$ be a finitely presented group. The rank of $G$ is defined to be the size of smallest generating set of $G$. The deficiency ${\rm def}(G)$ of $G$ is defined to be the maximum ...

**10**

votes

**1**answer

377 views

### Kazhdan's property T for Kahler surfaces

Is it true that the fundamental groups of compact Kahler surfaces have property T if and only if it they are finite? I am having trouble finding counterexamples to this, but maybe that's just me...

**6**

votes

**2**answers

281 views

### Groups quasi-isometric to reducible nonuniform lattices

It is known that a finitely group $G$ is quasi-isometric to a nonuniform irreducible lattice $\Lambda$ in a semisimple Lie group if and only if $G$ and
$\Lambda$ are commensurable (see references in ...

**4**

votes

**1**answer

432 views

### Dehn Twist in the sense of Geometric Group Theory and a Graph of Groups

Hello!
Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or a subgroup of a group. I ...

**3**

votes

**1**answer

349 views

### Relationship between hyperbolicity in group theory and hyperbolicity in geometry

Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional ...

**17**

votes

**1**answer

453 views

### Is the property of not containing $\mathbb{F}_2$ invariant under quasi-isometry?

Is the property of not containing the free group on two generators invariant under quasi-isometry? Amenability is, so if there is a counterexample it is also a solution to the von Neumann-Day ...

**5**

votes

**1**answer

261 views

### Flat sector in a proper cocompact CAT(0) space

Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 ...

**11**

votes

**3**answers

2k views

### Examples of “Monster” groups

I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are:
1.) ...

**7**

votes

**5**answers

857 views

### Analogues of the dihedral group

A virtually-$\mathbb{Z}$ group $G$ admits either a epimorphism onto $\mathbb{Z}$ or a epimorphism onto $D_\infty$.
So what happens if one replaces $\mathbb{Z}$ by another group $F$ (like the free ...

**15**

votes

**3**answers

1k views

### The second homotopy group of a simple CW-complex

Let $X$ be a CW-complex with
one 0-cell
two 1-cells
three 2-cells
no cells in dimensions 3 or higher.
Is it always true that $\pi_2(X)\ne 1$?

**4**

votes

**1**answer

380 views

### Examples of CAT(0)-groups

My question is the following:
Let M be a simply connected Riemannian manifold whose sectional curvatures
are all nonpositive and let G be a group. Suppose that G acts in M properly discontinuous and
...

**10**

votes

**2**answers

473 views

### Centralizers of non-iwip elements of $Out(F_n)$

Does there exist an infinite order element $\phi\in Out(F_n)$, for some or all $n\geq 3$, which is not iwip but has finite index in its centralizer? How about an element such that all its non-zero ...

**6**

votes

**3**answers

482 views

### Are subgroups of hyperbolic groups quasiisometrically embedded ?

Given a finitely generated subgroup of a finitely generated hyperbolic group. Is it true that the inclusion of each subgroup is a quasiisometric embedding ?
The first example for a group that does ...

**4**

votes

**1**answer

242 views

### Asymptotic dimension of graph manifold groups

Does every non-geometric graph manifold have fundamental group of asymptotic dimension 3?
This is affirmed in http://arxiv.org/abs/0909.1098 for closed graph manifolds, but I am interested in ...

**8**

votes

**0**answers

343 views

### is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...

**5**

votes

**1**answer

358 views

### Growth of groups versus Schreier graphs

This question is motivated by this one What is the relation between the number syntactic congruence classes, and the number of Nerode relation classes? where it essentially asks to compare the growth ...

**0**

votes

**1**answer

279 views

### Hyperbolicity of a fundamental group

Let G be the fundamental group of a compact 3-manifold which supports on its interior a complete non positively curved Riemannian metric and is a cilinder near de metric. Is G hyperbolic?

**4**

votes

**1**answer

237 views

### Hyperbolicity of a semidirect product

Let F be a finitely generated free group and let $\gamma : F \rightarrow F$ be an automorphism. Is the semidirect product $F \rtimes \mathbb{Z}$ an hyperbolic group? where $\mathbb{Z}$ acts in F via ...

**18**

votes

**3**answers

813 views

### Is there a non-Hopfian lacunary hyperbolic group?

The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...

**3**

votes

**1**answer

361 views

### Finite quotients of graphs

If $X$ is an infinite graph, $G$ is a group acting on $X$ with finite quotient; make $Y=X/G$ into graph of groups by attaching stabilizers at vertices and edges. Let $Z$ be a graph of groups, with ...

**4**

votes

**1**answer

471 views

### Groups acting on graph

Let $S$ be an infinite graph, $G$ is a group acting (effectively) on $S$ with finite quotient graph $S/G$. Make $S/G$ into graph of groups in obvious way by assigning stabilizers at vertices and ...

**8**

votes

**1**answer

607 views

### Surface groups and non separating loops

QUESTION: Let $g \geq 4$, $S(g)$ be the fundamental group of the genus $g$ surface, and $G$ be finitely generated (the number of generators $\leq 3$) group with abelianization of rank less than equal ...

**1**

vote

**3**answers

524 views

### Serre for complexes of groups

A theorem of Serre states that any finite subgroup $F$ of an (injective) amalgam $G_1 *_H G_2$ is conjugate into one of the factors $G_1$ or $G_2$, that is, $F$ is a subgroup of a vertex stabilizer of ...

**7**

votes

**3**answers

438 views

### Embedding groups into groups with some vanishing homology groups

Which finite subsets $S \subset \mathbb{N}$ have the following property : every countable group $G$ embeds into a finitely generated group $\Gamma$ such that $H_i(\Gamma;\mathbb{Z})=0$ for all $i \in ...

**5**

votes

**2**answers

707 views

### Residual Finiteness of Fundamental Group of Compact 3-Manifold

Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi_1(M)$ is residually finite. The outline of the proof is basically:
...

**6**

votes

**3**answers

529 views

### Braid group analogue for signed symmetric group?

This is probably something well known (either in the affirmative or in the negative) but I couldn't get this information easily:
Braid group:Symmetric group::?:Signed symmetric group
By "signed ...

**24**

votes

**4**answers

1k views

### Trees in groups of exponential growth

Question: Let $G$ be a finitely generated group with exponential growth.
Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree?
...

**5**

votes

**1**answer

456 views

### Percolation in Cayley graphs of semigroups.

Percolation in Cayley graphs of groups are studied by many researchers. There are also the concept Cayley graphs for semigroups. Are there any research about percolation in Cayley graphs for ...

**5**

votes

**2**answers

797 views

### Quasi-isometries vs Cayley Graphs

The following questions might be trivial, however, I couldn't solve them:
Let $G$ be generated by a finite symmetric set $S.$ Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of ...