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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4
votes
1answer
165 views

Periodic automorphisms of free groups and surface homeomorphisms

Denote by $F_n$ the free group of rank $n$. We say that an automorphism $\phi\in Aut(F_n)$ is geometric if there exists a surface with boundary $M$ and a homeomorphism $h\colon M\to M$ such that $h$ ...
8
votes
2answers
645 views

Actions of Thompson group F

Does anybody know the actions of Thompson group F which are not conjugate to the standard one? Motivation is to find actions such that the Schreier graph of the action does not contain a binary tree. ...
7
votes
1answer
233 views

Unbounded metrics on groups

If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?
2
votes
1answer
122 views

Polynomial growth of the Betti number of balls of the Cayley graphs

Consider a finitely generated group. Assume that the first Betti number of the ball of radius n in the Cayley graph is at most polynomial in n. This property is satisfied by free groups and groups of ...
5
votes
0answers
163 views

Can one characterize amenable groups with $C_G(x)$ cyclic for all $x\neq 1$?

Can one characterize the amenable groups that have the property that the centralizer of every non-identity element is cyclic? For example, must they be solvable?
4
votes
2answers
265 views

Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators?

We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set. I have been constructing a space ...
5
votes
1answer
313 views

Thompson's group T

Does there exist a non trivial homomorphism from Thompson's group T to a linear group?
0
votes
1answer
226 views

Ordered groups - examples

Let $G=BS(m,n)$ denote the Baumslag–Solitar groups defined by the presentation $\langle a,b: b^m a=a b^n\rangle$. We assume that G is non-abelian. Question: Find $m,n$ such that $G$ is an ...
13
votes
2answers
497 views

Normal subgroups of finite index in free groups

Hi all, This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of all normal subgroups ...
3
votes
0answers
93 views

Cogrowth and value of its series at the critical exponent

Let $G$ be a finitely generated group and write $G = F/N$ for $N$ a normal subgroup of a free group $F$. Let $S_n$ be the elements in $F$ written as words of exactly $n$ letters. So, for $n\geq 1$, ...
5
votes
2answers
419 views

Are virtual cubulated groups cubulated?

Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex? Edit: After ...
1
vote
1answer
384 views

Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...
5
votes
1answer
347 views

Classification of geometric outer automorphisms of free groups

Good evening everyone, an outer automorphism $[\phi]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism $h\colon M\stackrel{\cong}{\to}M$, where $M$ is a compact surface with ...
10
votes
0answers
502 views

Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
7
votes
1answer
242 views

How large is this “algebra” of defining graphs for Right-angled Artin groups?

As part of my research, I have been trying to construct a spherical space at infinity for every right-angled artin group. I've been able to work it out for a certain class of defining graphs. I'd like ...
3
votes
1answer
224 views

On Canonical generators of torsion free nilpotent group

I read somewhere that Maltcev proved that for any finitely generated torsion free Nilpotent group $G$ there are canonical generators, i.e. $g_{1},\ldots,g_{k}$ such that any $g \in G$ can be written ...
7
votes
0answers
361 views

Is it true that every f.g. infinite simple group has exponential growth?

Is it true that every finitely generated infinite simple group has exponential (word-)growth? Remark: As Mark Sapir has pointed out, the question whether every finitely generated group of ...
7
votes
4answers
403 views

isometric embeddings of Cayley graphs in “nice” spaces

This is from a physicist I know and as may be expected, I am threading my way between poorly defined and poorly translated. What groups have Cayley graphs (w.r.t. a fixed finite generating set, and ...
7
votes
1answer
177 views

Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups

I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions. $G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ ...
8
votes
1answer
170 views

Do finitely generated groups of polynomial growth satisfy a “uniform covering property?”

Let $G$ be a finitely generated discrete group with a finite symmetric generating set $S=S^{-1}\subset G$. For every group element $g$, define $\|g\|_S$ to be the length with respect to $S$, i.e. the ...
4
votes
1answer
406 views

Recognizing the stablizer of a degenerate three forms in six dimension

Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{*}\Omega=\Omega$ }. we say three-form $\Omega\in\wedge^{3}V^{*}$ is non-degenerate , if $i_X\Omega\neq 0$ for all $X\in V$-{0} Let $V\cong ...
5
votes
2answers
569 views

When is a Baumslag-Solitar group linear?

The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation $BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!
2
votes
1answer
417 views

why the group $GL(6,V)$ has an open orbit?

N.Hitchen in his paper about geometry of three forms wrote that "for a Real vector space $V$ of dimension six, the group $GL(6,V)$ has an open orbit and he referenced it to a thesis which was written ...
5
votes
1answer
205 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
1answer
189 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
2answers
352 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
2answers
232 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
3answers
278 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
2answers
238 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
0answers
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 ...
10
votes
6answers
911 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
1answer
328 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
2answers
225 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
1answer
188 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
2answers
330 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
0answers
203 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
1answer
544 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
1answer
631 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
2answers
820 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
2answers
167 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
2answers
482 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
2answers
351 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
3answers
431 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 ...
6
votes
1answer
200 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
1answer
713 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
1answer
363 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
2answers
274 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
1answer
393 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
1answer
339 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
1answer
440 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 ...