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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5
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0answers
204 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
550 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
634 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
829 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
169 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
485 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
359 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
441 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
202 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
714 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
367 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
275 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
404 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
343 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
445 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
1answer
243 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 ...
10
votes
3answers
1k 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.) ...
6
votes
5answers
743 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
3answers
966 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
1answer
328 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
2answers
449 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
3answers
419 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
1answer
233 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
0answers
304 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
1answer
329 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
1answer
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
1answer
228 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
3answers
752 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
1answer
357 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
1answer
425 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
1answer
597 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
3answers
505 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
3answers
399 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
2answers
667 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
3answers
511 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 ...
23
votes
4answers
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
1answer
453 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
2answers
717 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 ...
17
votes
4answers
777 views

When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?

The question has an easy answer, if one replaces free by free abelian: Then the resulting group is always solvable and a solvable subgroup of a CAT(0) group is virtually abelian. If the resulting was ...
5
votes
2answers
396 views

Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?

Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely ...
10
votes
4answers
897 views

What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the most general setting for which it might expected to be true?

What I would like to know is exactly what the title asks: What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the ...
24
votes
2answers
1k views

Invertible matrices satisfying $[x,y,y]=x$.

I have been thinking about this question for quite some time but now this question by Denis Serre revived some hope. Question. Let $x,y$ be invertible matrices (say, over $\mathbb C$) and ...
14
votes
1answer
545 views

Loop spaces and infinite braids

The Artin braid groups $B_n$ and the symmetric groups $S_n$ are closely related by the maps $1 \to P_n \to B_n \to S_n \to 1$. The infinite symmetric group has interesting interactions with homotopy ...
2
votes
1answer
327 views

balls as Foelner sets

This is essentially equivalent to this question by Simon Thomas. Let $G=\langle X\rangle$ be a finitely generated group, $b_n$ be the number of elements in the ball of radius $n$ in the Cayley ...
15
votes
9answers
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?
7
votes
2answers
348 views

Infinite loop space maps into or out of BAut(F_n)

There is an inclusion $S_n \to Aut(F_n)$ from the symmetric group into the automorphism group of a free group. After applying the Quillen +-constriction, both $BS_{\infty}$ and $BF_{\infty}$ become ...
10
votes
1answer
433 views

Partial word orders on groups

This is a followup question related to this question. Recall that a left-invariant partial order on a finitely generated group $G$ is called a partial word order if for every $a\le b\le c$ we have ...
6
votes
2answers
283 views

orders and length functions on finitely generated groups

Let $G$ be a finitely generated group with the natural word length function ($|x|$ is the length of the shortest word in generators of $G$ representing $x$). We call a partial left invariant order ...
14
votes
4answers
1k views

Braid groups acting on CAT(0)-complexes

Does the braid group $B_n, n\ge 3$, act properly by isometries on a CAT(0) cube complex? Update 1. During a recent talk of Nigel Higson in Pennstate Dmitri Burago asked whether the braid groups are ...
7
votes
2answers
369 views

Residually finite + torsion free + finite index = finite complex?

Suppose $G$ is a residually-finite group and $H < G$ a torsion-free subgroup of finite index. What characterizes such $G$ such that $BH$ is homotopic to a finite complex? I believe Serre ...