# Questions tagged [combinatorial-group-theory]

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155
questions

6
votes

1
answer

449
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### Are Artin-Tits groups ordered groups?

We consider Artin-Tits groups of two generators $(I_2(n))$. Are these groups ordered groups?

3
votes

1
answer

146
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 ...

5
votes

1
answer

257
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### 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=\...

3
votes

0
answers

454
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### Show me that I have not simplified the proof of the Adian-Rabin theorem

I am not a mathematics researcher but I am concerned that this question, posed with slightly different wording on math.stackexchange, may be too esoteric for that forum since it concerns the details ...

2
votes

1
answer

334
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### Proving certain triangle groups are infinite

[Cross-posted from MSE]
Consider the Von Dyck group
$$ G = \langle x,y\mid x^a=y^b=(xy)^c=1\rangle $$
where $a,b,c\ge3$. Because $G$ is infinite and residually finite, it has an infinite family of ...

2
votes

0
answers

58
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### 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 ...

2
votes

0
answers

150
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### The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$

Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...

3
votes

0
answers

222
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### What is the latest progress on the Andrews-Curtis Conjecture?

Out of curiosity . . .
What is the latest progress on the Andrews-Curtis Conjecture?
What's available online seems limited. (See the Wikipedia article linked to above.)
I found the following here:
...

18
votes

1
answer

728
views

### Is solvability semi-decidable?

Let $G = \langle A \mid R \rangle$ be a finitely presented group, given by a finite presentation. If $G$ is abelian, then we can verify this fact: simply verify the fact that $[a, b] = 1$ for all ...

3
votes

2
answers

166
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### Subsets of free groups contained in $2$-generated subgroups

$\DeclareMathOperator\rank{rank}$Let $F$ be a non-cyclic free group.
For which finitely generated subgroups $H< F$ such that $H$ is not of finite index in a free factor of $F$ does there exist a ...

16
votes

1
answer

802
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### 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 ...

7
votes

1
answer

556
views

### Do cyclically presented groups of positive word length four relators satisfy the Tits Alternative?

I finished an MPhil a year ago that focused on the following question. I've moved on to a different area of group theory now, so I thought I'd ask it here.
Definition: Let $w\in F_n$ for the free ...

5
votes

1
answer

187
views

### Can hyperbolic surfaces approximate every connected compact metric space?

Let $X$ be a connected compact metric space.
Question: Is there a sequence of compact hyperbolic surfaces (the curvature may differ between surfaces) that converges to $X$ in the Gromov-Hausdorff ...

1
vote

0
answers

74
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### Cohomological finiteness (boundedness) property

Let $G$ be arbitrary group. Let us assume it is $\operatorname{FP}_\infty$. Suppose that the integral cohomology groups $H^i(G, \mathbb{Z})$ have bounded rank as finitely generated free abelian groups ...

2
votes

1
answer

225
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 ...

8
votes

2
answers

469
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### 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 ...

4
votes

2
answers

218
views

### Presentationally finite group "extensions"

Fix a group $G$ and fix a presentation of $G$ as $\langle X\mid R\rangle$. A presentationally finite extension of $G$ is any group that can be presented as $H=\langle X\cup X'\mid R\cup R'\rangle$, ...

5
votes

0
answers

191
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### Finite groups with number of generators strictly less than number of relations

For the finite cyclic group of order $n$, there is the standard presentation $\langle a \mid a^n\rangle$. Also for $S_n$ (symmetric group), I know a few presentations where the number of relations is ...

3
votes

0
answers

127
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### 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_{\...

4
votes

0
answers

200
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### 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 $\phi$...

6
votes

2
answers

538
views

### Is It possible to determine whether the given finitely presented group is residually finite with MAGMA or GAP?

I am working on finitely presented groups with more than 5 generators and relators and I'm so curious: is it possible to determine residually finitness of finitely presented groups with MAGMA or GAP?

3
votes

2
answers

187
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### 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 ...

10
votes

2
answers

783
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### 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 ...

17
votes

3
answers

1k
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### 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 (...

2
votes

1
answer

203
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 ...

5
votes

1
answer

377
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 ...

5
votes

0
answers

132
views

### Can we define partial group actions on (finite) sets via generators and relators?

Let $G = \langle Y | R \rangle$ be a finitely presented group. A partial group action on a set $X$ is a premorphism into the inverse semigroup
$$
\mathcal I (X) = \{ f: A \to B : A, B \subseteq X, f\...

4
votes

1
answer

282
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 ...

1
vote

1
answer

242
views

### Which properties can be read off the balls of a Cayley graph?

For which properties (P) [of groups] does the following hold:
given a group $G$ which has a finite presentation with at most $n$ relations of length at most $\ell$, there is a $R(n,\ell)$ so that, if ...

8
votes

1
answer

354
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?"
...

9
votes

1
answer

367
views

### Finite presentability of semi-direct product of free group and its commutator subgroup

Let $F_n$ be a free group of rank $n \geq 2$. The group $F_n$ acts on its commutator subgroup $[F_n,\, F_n]$ by conjugation. Let $G = [F_n,\, F_n] \rtimes F_n$. It's not hard to see that $G$ is ...

7
votes

0
answers

177
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### Completeness of automorphism groups of free metabelian groups

I am not very familiar with free metabelian groups, so I apologise in advance if this is trivial.
A group $G$ is said to be complete if every automorphism of $G$ is inner. In this case, $\operatorname{...

7
votes

0
answers

269
views

### Uniform word problem in finitely presented simple groups

The following question arose in the comments on this question, and it seems like a reasonable question to ask in its own right. I've added some additional details.
The word problem in any fixed ...

9
votes

3
answers

581
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### 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 ...

12
votes

1
answer

387
views

### Commutator problem vs conjugacy/word problem

For a finitely presented group $G$, generated by a finite set $A$, the commutator problem is the decision problem: given a word $w$ over the alphabet $A \cup A^{-1}$, can one decide if $w$ is a ...

9
votes

1
answer

276
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 ...

7
votes

1
answer

207
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### Howson property of automorphism group of $F_2$ and of $F_3$

Is the intersection of any two finitely generated subgroups of $\operatorname{Aut}(F_2)$ (resp. $\operatorname{Aut}(F_3)$) again finitely generated? That is, does $\operatorname{Aut}(F_2)$ (resp. $\...

8
votes

0
answers

114
views

### The conjugacy problem for two-relator groups

Is the conjugacy problem for two-relator groups known to be undecidable?
The word problem for two-relator groups is a famous open problem (appearing e.g. as Question 9.29 in the Kourovka notebook), ...

0
votes

1
answer

177
views

### Examples of infinitely presented non-LEF groups

A group is LEF (locally embeddable in the class of finite groups) if it embeds into an ultraproduct of finite groups. Residually finite groups are LEF and finitely presented LEF groups are residually ...

5
votes

0
answers

196
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### 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 ...

2
votes

2
answers

287
views

### Combinatorial problem in $G(32, \, 6)$

The following problem arose when studying the same type of questions in Algebraic Geometry that led me to my previous question MO379272.
Let us consider the group $G$ of order $32$ whose label in GAP4 ...

4
votes

1
answer

130
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### Can one reduce to 'reversing' the right multiplier finite-state automata of an automatic group to obtain a biautomatic structure?

Let $\left( G, A, W, \left\{ R_{a} \right\}_{a \in A \cup \{ 1 \}} \right)$ be a group equipped with an automatic structure, where $G$ is the group, $A$ is a finite set of generators of $G$, $W$ is ...

2
votes

1
answer

624
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### Combinatorial problem in $\mathsf{S}_4$

I am working on a problem in Combinatorial Group Theory related to a construction in Algebraic Geometry, and I would like to have a conceptual proof of the fact described below.
I am looking for ...

2
votes

1
answer

229
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### Proving an inequality regarding number of transitive subgroups of the symmetric group

I defined the sequence $t$ where where $t(n)$ is the number of transitive subgroups of $S_n$ where we regard conjugate subgroups as distinct, i.e. the labeled version of A002106 at the OEIS.
Then I ...

18
votes

1
answer

549
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.
...

4
votes

2
answers

222
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### CCT groups of order $\leq 32$

A finite, non-abelian group $G$ is said to be a center
commutative-transitive group
$($or a CCT-group, for short$)$ if commutativity is a transitive
relation on the set on non-central elements. In ...

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{...

3
votes

1
answer

240
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### 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 ...

0
votes

0
answers

125
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### Intersection of subgroup of a free group with the lower central series

If I have a subgroup $S$ of a free group $\mathcal{F}_m$, what can I say about the behaviour of the descending sequence of subgroups
$\left< S, \Gamma_c(\mathcal{F}_m) \right>$ (where $\Gamma_c(\...

10
votes

2
answers

385
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### Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups?

I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator ...