**18**

votes

**0**answers

546 views

### Reference request: Parallel processor theorem of William Thurston

Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ...

**16**

votes

**0**answers

518 views

### How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...

**13**

votes

**0**answers

307 views

### Linear groups which don't contain products of free groups

Let $G \subset GL(n, \Bbb Z)$ be a f.g. linear group. The Tits alternative says that $G$ is either virtually solvable (i.e. has a solvable subgroup of finite index), or contains a free group $F_2$. ...

**13**

votes

**0**answers

234 views

### Bounding the lengths of the conjugators in the word problem for finite group presentations

Let $G = \langle X \mid R \rangle$ be a group defined by a finite presentation, and let $F$ be the free group on $X$. If $w \in F$ represents the identity in $G$, then $w$ is equal in $F$ to (the free ...

**12**

votes

**0**answers

318 views

### What is an example of a word hyperbolic group without a finite complete rewriting system?

I believe that it was an open question back when I was a graduate student whether every word hyperbolic group admits a finite complete (=Church-Rosser=Noetherian+confluent) rewriting system for some ...

**12**

votes

**0**answers

345 views

### Splay trees and Thompson's group $F$

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but:
Question: Is there a reformulation of the Dynamic ...

**10**

votes

**0**answers

547 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 ...

**9**

votes

**0**answers

248 views

### Group with unsolvable conjugacy problem but solvable conjugacy length?

Could there exist a finitely presented group with unsolvable conjugacy problem, in which it is decidable whether a word over the group generators is a shortest representative of an element in its ...

**8**

votes

**0**answers

199 views

### A Magnus theorem in the category of residually finite groups

There is a natural notion of a presentations in the category of residually finite groups. Namely, if $X$ is set and $R$ is a set of words in the free group $FG(X)$ on $X$, then define $G=RF\langle ...

**8**

votes

**0**answers

434 views

### Uniquely geodesic groups

Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...

**8**

votes

**0**answers

416 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 ...

**8**

votes

**0**answers

361 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 ...

**7**

votes

**0**answers

166 views

### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that:
For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$
contains infinitely many finite index ...

**7**

votes

**0**answers

82 views

### Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...

**6**

votes

**0**answers

202 views

### Polynomial growth without Gromov's theorem

It is a consequence of the theorem of M. Gromov on groups with polynomial growth and a result of P. Pansu(*) that if a finitely generated group $\Gamma = \langle S\rangle,\, S=S^{-1}$ satisfies ...

**6**

votes

**0**answers

145 views

### CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...

**6**

votes

**0**answers

143 views

### Homology groups of Noetherian groups

Let $G$ be a Noetherian group. Is $H_n(G,\mathbb{Z})$ finitely generated? Do we know the above for the special cases $n=2,3$ even?

**6**

votes

**0**answers

165 views

### “Twisted” Lyndon equation in a free group

In 1959, Lyndon showed that in a free group, the equation $u^2v^2=w^2$ has only commuting solutions: $uv=vu=w$. Is there in the litterature any information about the following "twisted" version of the ...

**6**

votes

**0**answers

110 views

### Two-relator products of cyclic groups

In "A proof of the Scott–Wiegold conjecture on free products of cyclic groups" Howie proved that every one-relator product of three cyclic groups is nontrivial. Is there a now proven theorem that says ...

**6**

votes

**0**answers

145 views

### How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...

**6**

votes

**0**answers

153 views

### Dynamics of virtual automorphisms of free group

The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism.
Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the ...

**5**

votes

**0**answers

121 views

### Generators of pure braid groups of arbitrary Coxeter groups

Let $W$ be an arbitrary Coxeter group, and let $A$ be the associated Artin-Tits braid group, with standard Coxeter generators $\sigma_i\in A$. Let $P$ be the "pure braid group", the kernel of the ...

**5**

votes

**0**answers

165 views

### Wild automorphisms of a free group

Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) ...

**5**

votes

**0**answers

254 views

### Quotients of an extension of the Higman group

(Note: this started as a different question that soon changed form, thanks to the answers.)
Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ ...

**5**

votes

**0**answers

108 views

### Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...

**5**

votes

**0**answers

117 views

### A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers

I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that
a) the graph $\Gamma$ is fine,
b) $\Gamma$ is not a tree,
c) not all ...

**5**

votes

**0**answers

82 views

### Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows:
Suppose ...

**5**

votes

**0**answers

95 views

### A subgroup of corank 1 in a free group contains a primitive element?

Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq ...

**5**

votes

**0**answers

245 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**5**

votes

**0**answers

123 views

### Centralizers in virtually special groups

Let $G$ be a virtually compact special group, in the terminology of Haglund and Wise (i.e., $G$ has a finite index subgroup $H$ which is isomorphic to the fundamental group of some compact special ...

**5**

votes

**0**answers

318 views

### Computing the pro-solvable closure of a finitely generated subgroup of a free group

The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...

**5**

votes

**0**answers

169 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?

**5**

votes

**0**answers

212 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 ...

**4**

votes

**0**answers

103 views

### Volume growth of balls

Let $G$ be a locally compact group and $K\subset G$ a compact subgroup. Suppose that on the homogeneous space $X=G/K$ we have a $G$-invariant proper metric $d$. For $R>0$ let $B(R)$ be the open ...

**4**

votes

**0**answers

77 views

### Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...

**4**

votes

**0**answers

77 views

### Roller's problem on median groups

At the end of his dissertation Poc Sets, Median Algebras and Group Actions, Martin Roller asks
A group $G$ is called median if it acts freely and transitively on a median algebra. This is ...

**4**

votes

**0**answers

102 views

### An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132)
For the argument to ...

**4**

votes

**0**answers

86 views

### Number of unitary representations of a Kazhdan group

It was proved by de la Harpe, Robertson, and Valette that for a discrete group $\Gamma$ with Kazhdan's property (T), there is a constant $c$ so that the number of irreducible unitary representations ...

**4**

votes

**0**answers

172 views

### Primitive elements in a free group

Let $F$ be a free group of rank $\aleph_0$ and let $H$ be a subgroup for which there exists some $a \in F$ such that $\langle H \cup \{a\} \rangle = F$. Must there be some free basis $B$ for $F$ and a ...

**4**

votes

**0**answers

113 views

### Is a finitely generated subgroup of a free profinite group virtually a retract?

Let $F$ be a nonabelian finitely generated free profinite group, and let $H \leq F$ be a finitely generated closed subgroup. Must there be some open subgroup $H \leq U \leq F$, and a closed normal ...

**4**

votes

**0**answers

93 views

### Actions of amenable groups on graphs with uncountably many ends

Let $G$ be a finitely generated amenable group acting transitively on an amenable Schreier graph $S$. Is it possible for $S$ to have uncountably many ends? An amenable graph with uncountably many ends ...

**4**

votes

**0**answers

83 views

### Nice minimal embeddings of large finite groups into compact Riemannian manifolds

The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...

**4**

votes

**0**answers

108 views

### Regularity of polynomial growth of groups

Let $G$ be a finitely generated group of polynomial growth. This means that the size $B_n$ of the ball of radius $n$ satsifies:
$$
A n^d \leq B_n \leq Bn^d
$$
for some constants $A$, $B$.
My question ...

**4**

votes

**0**answers

127 views

### Uniform incentre of collection of quasi convex subspaces in hyperbolic spaces

I'm reading a paper of Wise on cubulations and the following fact is used:
Let $H$ be a quasi-convex subgroup of a $\delta$-hyperbolic group and let $H_i, i\in I$ be a finite family of translates of ...

**4**

votes

**0**answers

239 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 ...

**3**

votes

**0**answers

75 views

### Invertibility of group Laplacian in $\ell^1$

Let $G$ be a discrete group and let $S$ be a generating set for $G$; assume that $S$ is symmetric (i.e., $g\in S$ iff $g^{-1}\in S$). Let $L=L_S=\frac{1}{|S|}(\sum_{g\in S} g-1)$ be an element of the ...

**3**

votes

**0**answers

108 views

### Short exact sequences for amalgamated free products and HNN Extensions

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I would post it here too:
If $A$ and $B$ are groups we have the following short exact sequence:
$$ 0 \to ...

**3**

votes

**0**answers

46 views

### Cubulating non-compact hyperbolic manifolds

Let $X$ be a hyperbolic manifold of arbitrary dimension. When does $X$ admit a cell structure of a CAT(-1) cube complex? of a hyperbolic CAT(0) cube complex?
I suspect that the question is widely ...

**3**

votes

**0**answers

66 views

### Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant?

Is the family of probabilities $\mu^n$ (convolution) generated by a random walk $\mu$ on a finitely generated amenable group $G$ asymptotically invariant ($\|g\mu^n-\mu\|_{L^1}\to 0$ for any $g\in ...

**3**

votes

**0**answers

161 views

### Uniform sub-linearity of sub-additive functions on groups

Suppose $G$ is a finitely generated group and suppose $f: G \to \mathbb{R}$ is subadditive function, that is: $f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such $f$ is the word length in some ...