Questions tagged [combinatorial-group-theory]

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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 ...
Jean Charles's user avatar
2 votes
1 answer
181 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 ...
Marcos's user avatar
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7 votes
2 answers
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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 ...
Carl-Fredrik Nyberg Brodda's user avatar
4 votes
2 answers
165 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 $H=\langle X\cup X'\mid R\cup R'\rangle$, ...
tomasz's user avatar
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5 votes
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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 ...
gola vat's user avatar
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3 votes
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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_{\...
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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 $\phi$...
ghc1997's user avatar
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6 votes
2 answers
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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?
ALan Kay's user avatar
3 votes
2 answers
160 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 ...
24601's user avatar
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10 votes
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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 ...
Jean Charles's user avatar
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3 answers
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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 (...
Jean Charles's user avatar
2 votes
1 answer
182 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 ...
Agelos's user avatar
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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 ...
ghc1997's user avatar
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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\...
jpmacmanus's user avatar
4 votes
1 answer
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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 ...
ghc1997's user avatar
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1 answer
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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 ...
ARG's user avatar
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8 votes
1 answer
293 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?" ...
Ross Griebenow's user avatar
9 votes
1 answer
319 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 ...
Cindy's user avatar
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7 votes
0 answers
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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{...
Carl-Fredrik Nyberg Brodda's user avatar
7 votes
0 answers
192 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 ...
Carl-Fredrik Nyberg Brodda's user avatar
9 votes
3 answers
421 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 ...
AGenevois's user avatar
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12 votes
1 answer
350 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 ...
Carl-Fredrik Nyberg Brodda's user avatar
9 votes
1 answer
269 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 ...
frafour's user avatar
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7 votes
1 answer
174 views

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. $\...
Carl-Fredrik Nyberg Brodda's user avatar
8 votes
0 answers
101 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), ...
Carl-Fredrik Nyberg Brodda's user avatar
0 votes
1 answer
147 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 ...
frafour's user avatar
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5 votes
0 answers
177 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 ...
Ethan Dlugie's user avatar
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2 votes
2 answers
277 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 ...
Francesco Polizzi's user avatar
4 votes
1 answer
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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 ...
user171576's user avatar
2 votes
1 answer
612 views

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 ...
Francesco Polizzi's user avatar
2 votes
1 answer
203 views

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 ...
John Erickson's user avatar
18 votes
1 answer
513 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. ...
Matt Zaremsky's user avatar
4 votes
2 answers
194 views

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 ...
Francesco Polizzi's user avatar
7 votes
0 answers
410 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{...
Francesco Polizzi's user avatar
3 votes
1 answer
203 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 ...
Siddharth Bhat's user avatar
0 votes
0 answers
97 views

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(\...
Thomas Meyer's user avatar
10 votes
2 answers
360 views

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 ...
jpmacmanus's user avatar
3 votes
1 answer
127 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 ...
user101010's user avatar
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2 votes
0 answers
123 views

Time complexity of randomized algorithm: right-multiplying by random elements $z_i$ from a group $H$ to achieve $H$-invariance

Note: This question was inspired by a related question about the Quantum Merlin Arthur (QMA) complexity class on Quantum Computing Stack Exchange. I was deliberating whether to ask this on CS Theory ...
Sanchayan Dutta's user avatar
5 votes
0 answers
175 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 $F$. ...
frafour's user avatar
  • 435
12 votes
1 answer
768 views

Minimum number of relations that must be added to make a group abelian

Let $G$ be a finitely generated group and let $c(G)$ denote the minimal number of relations that we must add to a presentation fro $G$ in order to make $G$ abelian. I would like to find examples of ...
user101010's user avatar
  • 5,299
4 votes
1 answer
156 views

Complementing the red and blue boolean cube?

Given a boolean $0/1$ cube in $n$ dimensions with $2^{n-1}$ red and $2^{n-1}$ blue points can we complement the cube (blue becomes red and vice versa) in $\operatorname{poly}(n)$ transformations? ...
Turbo's user avatar
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1 vote
0 answers
20 views

Do small subsets of $S_n$ subgroups cover almost all permutation configurations of $S_n$?

Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$. For ...
Turbo's user avatar
  • 13.2k
1 vote
0 answers
91 views

Are almost all permutation configurations from $S_n$ covered by small subsets subgroups of $S_n$?

Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$. Do we have ...
Turbo's user avatar
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15 votes
2 answers
896 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 ...
Sam Hughes's user avatar
8 votes
1 answer
216 views

Number of words of length N that reduce to the identity in a specific Coxeter group

Suppose we have a Coxeter group whose diagram is given by a simplex. In other words, $G=\langle g_1,\ldots ,g_k\mid(g_i)^2=e,\,(g_ig_j)^3=e \rangle$. How many words of length $N$ simplify to the ...
Craig's user avatar
  • 485
6 votes
0 answers
480 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 ...
Carl-Fredrik Nyberg Brodda's user avatar
9 votes
0 answers
267 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'...
Sebastien Palcoux's user avatar
6 votes
0 answers
471 views

Darkness in the lamplighter group

Consider paths through the lamplighter group $\mathbb{Z}_n\wr\mathbb{Z}$ with steps consisting of moving left, moving right, and toggling the lamp at the current position. How many paths of length $m$ ...
user76284's user avatar
  • 1,671
4 votes
3 answers
234 views

Conjugacy in right-angled Artin groups

I am looking for a reference containing the following result: Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the ...
AGenevois's user avatar
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