**23**

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

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

**2**answers

746 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

**4**answers

792 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

**2**answers

398 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

**4**answers

917 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

**2**answers

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

**1**answer

550 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

**1**answer

332 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

**9**answers

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

**2**answers

352 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

**1**answer

435 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

**2**answers

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

**4**answers

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

**2**answers

371 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

**3**answers

886 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

**2**answers

636 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

**2**answers

402 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

**1**answer

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

**4**answers

950 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

**1**answer

480 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

**0**answers

232 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

**3**answers

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

**2**answers

753 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

**4**answers

785 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

**2**answers

460 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

**4**answers

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

**1**answer

311 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 ...

**7**

votes

**2**answers

285 views

### Automorphisms of supergroups of non-coHopfian groups

In this question, I asked whether there existed groups $G$ with finitely presentable subgroups $H$ such that $gHg^{-1}$ is a proper subgroup of $H$ for some $g \in G$. Robin Chapman pointed out that ...

**8**

votes

**1**answer

442 views

### Conjugating a subgroup of a group into a proper subgroup of itself

The following question came up in the class I'm teaching right now. There definitely exist groups $G$ with subgroups $H$ such that there exists some $g \in G$ such that $g H g^{-1}$ is a proper ...

**11**

votes

**1**answer

807 views

### Topological HNN extensions

First, let me recall what an abstract HNN extension is. Let $G$ be an abstract group, $A, B < G$ be subgroups of $G$ and $\phi : A \to B$ be an isomorphisms. Then there is a group $H$ and an ...

**1**

vote

**1**answer

217 views

### blowing up the graphs

I heard the phrase from many mathematician using in the colloquials. I heard algebraic geometer using it. I was never bother about it until one of my professor responded to one my question as follows:
...

**5**

votes

**1**answer

252 views

### Connections between properties of a group and local symmetries of its Cayley graph

Hi everyone,
Let $\Gamma$ be a finitly generated group.
Does someone know of a connection between properties of $\Gamma$ to local symmetries of its Cayley graph?
More specificly, what can one learn ...

**4**

votes

**2**answers

441 views

### HNN extensions which are free products

which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...

**3**

votes

**1**answer

322 views

### How do Dehn functions of special linear and mapping class groups behave?

Hi,
I apologize for the basic questions. I am looking for good references on the following problems:
1) What is known about the Dehn function of $SL_n(\mathbb{Z})$?
2) What is known about the Dehn ...

**24**

votes

**9**answers

5k views

### Introductory text on geometric group theory?

Can someone indicate me a good introductory text on geometric group theory?