All Questions
Tagged with geometric-group-theory combinatorial-group-theory
56 questions
24
votes
4
answers
2k
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 it ...
- 25k
21
votes
1
answer
831
views
Can a hyperbolic, one ended, one relator group, have a shorter trivial word?
Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the ...
user35370
18
votes
1
answer
567
views
Is Thompson's group $T$ co-Hopfian?
A group $G$ is co-Hopfian if every injective homomorphism $G\to G$ is bijective, i.e., if $G$ contains no proper subgroups isomorphic to $G$. My question is whether Thompson's group $T$ is co-Hopfian.
...
- 3,740
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
16
votes
1
answer
850
views
A "simpler" description of the automorphism group of the lamplighter group
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references.
The lamplighter group is defined by the ...
- 823
15
votes
2
answers
1k
views
Is an HNN extension of a virtually torsion-free group virtually torsion-free?
This is a cross post from Math.StackExchange after 2 weeks without an answer and a bounty being placed on the question.
Let $G=\langle X\ |\ R\rangle$ be a (finitely presented) virtually torsion-free ...
- 855
15
votes
2
answers
2k
views
Dehn's algorithm for word problem for surface groups
For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...
- 153
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
13
votes
1
answer
199
views
Minimal length presentations of cyclic groups
By the length of a finite presentation I mean the sum of the lengths of the relators. I am interested in knowing what the minimal length of a presentation of $\mathbb{Z}/n\mathbb{Z}$. I'm even more ...
- 5,349
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
3
answers
666
views
Subgroups of RAAGs vs. subgroups of RACGs
Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group?
It is well-known from the theory of special cube complexes that ...
- 8,401
10
votes
2
answers
853
views
Examples of hyperbolic groups with non-hyperbolic subgroups
In a previous question, I asked about hyperbolic groups in which every finitely generated subgroup is hyperbolic. I am now curious about the reverse question: what are some examples of hyperbolic ...
- 383
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
2
answers
454
views
Minimal normally generating subsets of minimal generating sets
Let $G$ be a finitely generated group. The weight $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called ...
- 770
10
votes
0
answers
214
views
2-generator subgroups of an Artin group of small type
Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...
- 765
9
votes
3
answers
842
views
Is there a one relator group with property (T)?
Is there a one-relator group with property (T)?
That is, is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup ...
- 11.3k
9
votes
1
answer
398
views
When are biautomatic groups hyperbolic?
This list of open problems from http://grouptheory.info/ includes the question:
"Is every biautomatic group which does not contain any $\mathbb{Z} \times \mathbb{Z}$ subgroups, hyperbolic?"
...
- 123
9
votes
1
answer
281
views
Largest Hopfian quotient
Let $\Gamma$ be a group, say finitely generated if it helps. Does $\Gamma$ admit a largest Hopfian quotient? That is, does there exist a Hopfian quotient $H$ of $\Gamma$, such that every surjective ...
- 435
9
votes
0
answers
310
views
Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups
A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
9
votes
0
answers
556
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 ...
- 37.4k
8
votes
2
answers
486
views
Subgroup membership problem in simple groups
Let $G$ be a finitely presented simple group. By Kuznetsov (1958), $G$ has decidable word problem. However, by Scott [1], $G$ may have undecidable conjugacy problem. Is anything known about other ...
8
votes
1
answer
349
views
Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
8
votes
1
answer
327
views
Does every generating set of the first homology group of a Cayley graph give rise to a presentation of its group?
Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph.
Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set ...
- 1,926
7
votes
1
answer
503
views
Are Artin-Tits groups ordered groups?
We consider Artin-Tits groups of two generators $(I_2(n))$. Are these groups ordered groups?
7
votes
0
answers
420
views
Are these two kernels isomorphic groups?
We have a finitely presented, infinite group $\mathsf{B}$,
coming from a geometric topology problem (it is the quotient of a braid group for a genus 2 surface).
It is generated by elements
\begin{...
- 66.3k
6
votes
1
answer
564
views
K-fellow traveler property and automatic structure
I have been reading several articles about automatic groups and metric spaces of negative curvature. However it is not clear for me the relationship between automatic groups, hyperbolcity and the k-...
- 61
6
votes
0
answers
646
views
Minimum Simple Burger-Mozes Type Group
Burger and Mozes constructed (Burger and Mozes - Lattices in products of trees) infinite, finitely presented, torsion-free simple groups which split as amalgams of two finitely generated free groups ...
5
votes
1
answer
145
views
Computations with conetypes of hyperbolic groups
I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group ...
- 329
5
votes
1
answer
386
views
Is this semi-direct product residually finite?
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can help me find a way to check the residual finiteness of this group.
Consider the ...
- 823
5
votes
1
answer
284
views
Word length in the surface groups
I want to know if there are some results about the title of this question.
Let $G$ be an orientable closed surface group with genus $n$ greater than 1. We know it has a canonical presentation.
$$G=\...
- 53
5
votes
1
answer
264
views
Bases of surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. A $2g$-tuple $(x_1,y_1, \dots,x_g,y_g) \in \Gamma_g^{2g}$ will be called a Surface Basis if we have the presentation $$\Gamma_g = \langle x_1, ...
- 11.3k
5
votes
0
answers
216
views
Tools for computing from group presentations
What are some tools -- either theoretical/by hand or algorithmic/by computer -- that are useful for doing computations in finitely presented groups?
In my particular case, I'm working with a finitely ...
- 1,277
5
votes
0
answers
192
views
Description of quasimorphisms of the free group
Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...
- 435
5
votes
0
answers
285
views
Any method to detect subgroup generated by a subset of the generators from its presentation
I have met the following problem. A group $G$ is given as follows
$G = \langle x,y,t| y^{-2}xy^2 = x,t^{-1}yt =y^2 ,t^{-1}xt = xy^{-1}xy\rangle$
Is the subgroup generated by $y$ and $t$ just the ...
- 1,598
4
votes
2
answers
337
views
A Karrass-Solitar theorem for surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma_g = \langle x_1, y_1, \dots, x_g, y_g \vert \prod_{i = 1}^{g}[x_i,y_i] = 1\rangle$$
Is there a ...
- 11.3k
4
votes
1
answer
264
views
Geometric content of area of a word in geometric group theory?
Where does the idea of 'area' come from in Geometric Group Theory? The wikipedia article states that this definition was 'inspired' from Riemannian geometry:
Gromov's proof was in large part informed ...
- 1,221
4
votes
2
answers
460
views
A question about generating set of groups and epimorphism
Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an ...
- 2,118
4
votes
1
answer
294
views
Permuting subgroups with the same finite index
Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ with polynomial growth. Let $H$ be a subgroup of $G$ with finite index $m$. Let $\phi$ be an ...
- 823
4
votes
0
answers
214
views
Does there exist a finitely generated, torsion group $G$ with a residually finite ascending HNN extension?
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can provide me some insight.
Let $G$ be a group with an injective endomorphism $\phi$...
- 823
4
votes
0
answers
191
views
Log-concavity of the growth function
Given a Cayley graph of a group $G$ with finite generating set $A$ and exponential growth. Let $S_n$ be the elements whose word length is exactly $n$.
$\textbf{Question:}$ Is $f(n) = |S_{2n}|$ a log-...
- 4,432
3
votes
2
answers
197
views
HNN decomposition of finite rank free group over infinite rank subgroups
It's a nice result of Swarup that whenever a free group $G$ splits as an HNN extension $G = J \ast_{H,t}$ with $H$ a finitely generated subgroup, there exist splittings $J = J_1 \ast J_2$ and $H = H_1 ...
- 302
3
votes
2
answers
287
views
Free subgroup of a quotient
Let $F$ be a free group on $x,y,z$. Fix $n>1$ (I am ready to assume that $n$ is large enough). Let $\mathcal{W}$ be the set of cyclically reduced words $w$ in $F$ where the letter $z$ appears at ...
- 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
163
views
Bounding the size of the conjugating elements given the Dehn function
I am learning a little bit about Dehn functions of group presentations and I came across a question that is probably pretty basic but that I was giving me trouble. I'll set some notation but ...
- 5,349
3
votes
0
answers
132
views
the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
- 823
3
votes
0
answers
421
views
Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...
- 11.3k
2
votes
1
answer
439
views
Quotient groups of the lower central series of a surface group
In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
Q. What are the quotient ...
- 66.3k
2
votes
1
answer
239
views
Quotient of an Artin group is an Artin group
I'm working on a problem about Artin groups, and to simplify this problem I want to take a quotient that allow us to go to an easier Artin group, but I'm not sure if the quotient is well defined. This ...
- 911
2
votes
1
answer
232
views
Examples of group families with solvable uniform word problem
I would like to know of any examples of families of groups that are known (or conjectured) to have a solvable uniform word problem, i.e. an algorithm that given a presentation $P$ of a group in the ...
- 1,926
2
votes
0
answers
60
views
upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups
Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
- 823