All Questions
61 questions
52
votes
14
answers
14k
views
Introductory text on geometric group theory?
Can someone indicate me a good introductory text on geometric group theory?
23
votes
2
answers
2k
views
Modern references on hyperbolic groups
Several good references dedicated to hyperbolic groups have been written until 1990, including:
Hyperbolic groups, written by M. Gromov.
Géométrie et théorie des groupes : les groupes hyperboliques ...
- 8,401
18
votes
0
answers
734
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 ...
- 11.3k
17
votes
3
answers
1k
views
Examples of locally hyperbolic groups
It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
- 383
17
votes
1
answer
459
views
Existence of a quasi-isometric residually finite group?
It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...
- 733
16
votes
3
answers
1k
views
Torsion subgroups of hyperbolic groups are finite?
Is it true that torsion subgroups of hyperbolic groups are finite?
I have a vague memory that this is true, perhaps due to Ol'shanskii, but have been struggling to find a reference.
(By a torsion ...
- 161
16
votes
3
answers
2k
views
Your favorite papers on geometric group theory
I would like to improve my "depth of understanding" in geometric group theory. So I am interested in short and accessible papers on subjects related to this field but which are not always ...
13
votes
3
answers
2k
views
Which groups are LERF?
A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
- 11.3k
12
votes
2
answers
3k
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.) Non-...
- 2,441
11
votes
1
answer
619
views
Analogues of the curve complex for Out(F)
Let $F$ be a finitely generated free group.
Question: Is there an authoritative survey of analogues of the curve complex for $\mathrm{Out}(F)$? If not, as seems likely, would a passing expert be ...
- 25k
11
votes
1
answer
250
views
Recognising the elements of the Grigorchuk group
The Grigorchuk group $\mathfrak{G}= \langle a,b,c,d \rangle$ is a group of automorphisms of the infinite rooted binary tree $\mathcal{T}_2$. Every element of $\mathfrak{G}$ can be represented by a ...
- 8,401
11
votes
0
answers
345
views
Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
- 1,996
11
votes
0
answers
379
views
Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
- 525
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
10
votes
2
answers
815
views
Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4
I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
- 201
10
votes
1
answer
602
views
hyperbolic quotient of hyperbolic group
I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...
- 37.4k
9
votes
1
answer
738
views
Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
- 291
9
votes
3
answers
548
views
Spectral radius of a finitely generated group
Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...
- 1,407
9
votes
1
answer
337
views
amenable + without $BS(m,n)$+finite $K(G,1)$implies virtually cyclic?
I heard from someone that the following problem is an open question.
(Open Problem 1)For a countable discrete group $G$, suppose it does not contain any Baumslag-Solitar subgroups $BS(m,n):=\langle x,...
- 1,528
9
votes
1
answer
495
views
Divergence of Groups and Metric Spaces
Several papers, including this and this claim that divergence of finitely generated groups and metric spaces have been introduced by Misha Gromov in his paper "Asymptotic invariants of infinite groups"...
user6976
9
votes
1
answer
230
views
Yang-Mills algebra and lower central series of surface groups
Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...
8
votes
2
answers
618
views
Relative/acylindrical hyperbolicity of free-by-cyclic groups
Is this statement true?
Let $\mathbb{F}$ denotes a finitely generated free group, $\Phi$ an automorphism of $\mathbb{F}$ and $\varphi$ its image in $\mathrm{Out}(\mathbb{F})$.
If $\varphi$ is ...
- 1,755
8
votes
1
answer
227
views
Non-finitely presented FP groups with cohomological dimension $2$
In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...
- 15.9k
8
votes
1
answer
200
views
For which planar topological spaces $Z$ does there exist a hyperbolic group $\Gamma$ with $\partial \Gamma \cong Z$?
Recall a topological space is called planar if it can be embedded in $S^2$. I'm interested in understanding hyperbolic groups with planar boundaries.
In [1], it is shown that if a one-ended hyperbolic ...
- 639
8
votes
2
answers
272
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 ...
- 2,371
7
votes
3
answers
2k
views
Bass-Serre theory textbook
I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? ...
- 422
7
votes
1
answer
697
views
Growth of Thompson's group $F$
EDIT(August 2013): I accepted Mark's answer as being the state of art- there are two relevant references, one in the answer and one in the comments. The minimal growth rate of $F$ remains unknown with ...
- 992
7
votes
0
answers
252
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 ...
- 11.3k
6
votes
1
answer
399
views
One-ended finitely presented subgroups of hyperbolic groups
In Hyperbolic groups (page 82), Gromov claims that, by a standard application of Thurston's method of geodesic (hyperbolic) simplices, it can be prove that a hyperbolic group contains finitely many ...
- 2,371
6
votes
2
answers
483
views
Products of elliptic isometries
A well-known property on groups acting on trees is:
Theorem: Let $T$ be a tree and $g,h \in \mathrm{Isom}(T)$ two elliptic isometries. If $\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$ then the ...
- 2,371
5
votes
1
answer
329
views
A hyperbolic group with a small profinite completion
Is there a finitely generated non-elementary word hyperbolic group the profinite completion of which is known (or conjectured) to be rather restricted, that is: abelian, pro-$p$, virtually prosolvable,...
- 11.3k
5
votes
1
answer
175
views
Growth of the word norm for elementary matrices in $\rm SL_3 (\mathbb{Z})$
This is a reference request, since the answer is probably well known, but I could not find it.
Given a finitely generated group $\Gamma$ with a generating set $S$, define the word norm $l = l_S : \...
- 1,163
5
votes
1
answer
884
views
solvable word problem without algorithm
Let $G$ be a finitely generated group. I wonder if there are examples where:
1) The word problem is known to be solvable in $G$ but there is no algorithm known.
2) The word problem is known to be ...
- 829
5
votes
0
answers
200
views
Virtual fibring of $\mathrm{Out}(F_2\times F_2)$
A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated.
I want ...
- 911
5
votes
0
answers
228
views
Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
5
votes
0
answers
169
views
In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
4
votes
3
answers
320
views
Examples of IF-groups
I have seen that several authors say that an infinite group $G$ is an IF-group (or has the IF-property) if every subgroup of infinite index in $G$ is free (for instance, see https://arxiv.org/pdf/1607....
- 11.3k
4
votes
2
answers
1k
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 $x$ ...
- 25.8k
4
votes
0
answers
177
views
Ping pong with parabolic isometries on Gromov hyperbolic spaces
For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
- 41
4
votes
0
answers
124
views
Abelian-by-cyclic subgroups of exponential growth solvable groups
I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
- 4,432
3
votes
1
answer
271
views
Passing to normal forms in graphs of groups
Given a word $w \in X^{\pm 1}$ representing an element of the free group $F(X)$ there is a (usually non-unique) sequence $w=w_0 \to w_1 \to \cdots \to w_r$ with $|w_i|>|w_{i+1}|$ where $w_r$ is the ...
- 1,033
3
votes
1
answer
292
views
Special linear groups over function fields
Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements.
What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...
- 11.3k
3
votes
1
answer
165
views
When the fundamental group of subgraph of groups embeds?
Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
- 1,338
3
votes
1
answer
141
views
The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic
Given a group $G$, suppose $G$ admits a non-elementary acylindrical action
on a Gromov hyperbolic space $S$.
I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically ...
- 199
3
votes
0
answers
393
views
What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
3
votes
0
answers
493
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 [...
- 721
3
votes
0
answers
209
views
Growth of the number of generators in hyperbolic groups
Let $G$ be an infinite hyperbolic group, and let us further assume that it is residually finite (or even LERF/GFERF) so that we have plenty of subgroups of finite index.
I would like to know if one ...
- 11.3k
3
votes
0
answers
156
views
Cancellations in products of two elements of a hyperbolic group
Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
- 637
2
votes
1
answer
210
views
Reference proving $\beta^{(2)}_1(G) \le d(G)-1$
I saw a paper that said it is well-known that for finitely generated group $G$: $\beta^{(2)}_1(G) \le d(G)-1$, but I can't find any reference proving it.
$d(G)$ denotes the minimal number of ...
- 197