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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3
votes
1answer
347 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
451 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
251 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
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.) ...
7
votes
5answers
840 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
989 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
366 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
458 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
461 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
240 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
327 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
343 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
233 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
786 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
358 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
450 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
600 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
516 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
420 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
692 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
520 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
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
455 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
763 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
805 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
400 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
982 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
557 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
334 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
355 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
440 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
284 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
375 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 ...
7
votes
3answers
898 views

A finite index subgroup of the Mapping Class Group

Let $G$ be the mapping class group of a closed surface $S_{g}$. Bestvina-Bromberg-Fujuwara http://front.math.ucdavis.edu/1006.1939 recently constructed a finite index subgroup $B$ of $G$ such that for ...
2
votes
2answers
647 views

Reference request for two-generator subgroups of a free group

According to B. Fine, G. Rosenberger, On restricted Gromov groups, Comm. Algebra 20 (1992) 2171--2181, Gromov proved the following in his long article introducing word-hyperbolic groups: Let $x$ ...
8
votes
2answers
405 views

Which groups have nice compactifications ?

Given a discrete group G. Is there a nice criterion to decide, whether there is a compact Hausdorff $G$- space X, that contains the discrete space $G$ as a subspace, such that the stabilizer of every ...
31
votes
1answer
2k views

orders of products of permutations

Let $p$ be a prime, $n\gg p$ not divisible by $p$ (say, $n>2^{2^p}$). Are there two permutations $a, b$ of the set $\{1,...,n\}$ which together act transitively on $\{1,2,...,n\}$ and such that all ...
5
votes
4answers
960 views

Higher-dimensional braid group?

Let $\Delta$ be 2-disk. Let $C(\Delta;n)$ be a configuration space. i.e.) $C(\Delta;n)= \lbrace (z_1,\ldots,z_n)\in \Delta\times\ldots\Delta | z_i\neq z_j ~\textrm{if}~ i\neq j \rbrace $ Then, it ...
4
votes
1answer
487 views

Infinite direct products and derived subgroups

Suppose $G_1, G_2, \dots, G_n, \dots$ are groups (I use countable sequences, though the question is also applicable for uncountable collections of groups). Suppose G is the unrestricted external ...
4
votes
0answers
233 views

Can one pose a Toeplitz index problem associated to a discrete group?

Before posing my question, let me provide a little background since the Wikipedia page on this stuff is sorely lacking. Let's start with the classical case of the Toeplitz index problem on the ...
19
votes
3answers
1k views

An example of a non-amenable exact group without free subgroups.

A countable discrete group $\Gamma$ is said to be exact if it admits an amenable action on some compact space. So clearly amenable groups are exact, but large familes of non-amenable groups are as ...
3
votes
2answers
765 views

Hausdorff Dimension of Cayley Graphs of Groups

I was wondering what has been done concerning the Hausdorff measure of the Cayley graphs of finitely generated countable groups. There are number of issues that would need to be dealt with: 1.) By ...
16
votes
4answers
804 views

Free splittings of one-relator groups

Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings. Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is ...
13
votes
2answers
467 views

The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
29
votes
4answers
2k views

Is there a simple proof that a group of linear growth is quasi-isometric to Z?

I proposed to a master's student to work, from the exercise in Ghys-de la Harpe's book, on the proof that a finitely generated group $G$ that is quasi-isometric to $\mathbb{Z}$ is virtually ...
9
votes
1answer
313 views

Decidability of conjugacy problem for finitely generated subgroups of free groups

The conjugacy problem for a free group $F_n$ on $n$ letters has an easy solution. Each element of $F_n$ is conjugate to a unique and easily computable "cyclically reduced element" (this means that if ...