**14**

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

341 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

424 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

278 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 ...

**13**

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

363 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

819 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

600 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

387 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

884 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

461 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 ...

**5**

votes

**0**answers

229 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 ...

**20**

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

726 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 ...

**15**

votes

**4**answers

740 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

435 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 ...

**26**

votes

**4**answers

1k 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

293 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 ...

**6**

votes

**2**answers

275 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 ...

**7**

votes

**1**answer

425 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

772 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

214 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

247 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

387 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

300 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 ...

**22**

votes

**9**answers

4k views

### Introductory text on geometric group theory?

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