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Questions tagged [combinatorial-group-theory]

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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?
navashree chanania's user avatar
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 ...
tomasz's user avatar
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5 votes
1 answer
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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=\...
keqiyehuopo's user avatar
3 votes
0 answers
454 views

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 ...
Perry Bleiberg's user avatar
2 votes
1 answer
334 views

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 ...
Steve D's user avatar
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2 votes
0 answers
58 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 ...
ghc1997's user avatar
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2 votes
0 answers
150 views

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 $n\in \mathbb{Z}$, $...
ghc1997's user avatar
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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: ...
Shaun's user avatar
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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 ...
Carl-Fredrik Nyberg Brodda's user avatar
3 votes
2 answers
166 views

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 ...
ADL's user avatar
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16 votes
1 answer
802 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 ...
ghc1997's user avatar
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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 ...
Shaun's user avatar
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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 ...
LeechLattice's user avatar
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1 vote
0 answers
74 views

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
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 ...
Marcos's user avatar
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8 votes
2 answers
469 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 ...
Carl-Fredrik Nyberg Brodda's user avatar
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 $H=\langle X\cup X'\mid R\cup R'\rangle$, ...
tomasz's user avatar
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5 votes
0 answers
191 views

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
0 answers
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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_{\...
ghc1997's user avatar
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4 votes
0 answers
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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
  • 783
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?
ALan Kay's user avatar
3 votes
2 answers
187 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
  • 302
10 votes
2 answers
783 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 ...
Jean Charles's user avatar
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 (...
Jean Charles's user avatar
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 ...
Agelos's user avatar
  • 1,896
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 ...
ghc1997's user avatar
  • 783
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\...
jpmacmanus's user avatar
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 ...
ghc1997's user avatar
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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 ...
ARG's user avatar
  • 4,352
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?" ...
Ross Griebenow's user avatar
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 ...
Cindy's user avatar
  • 93
7 votes
0 answers
177 views

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
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 ...
Carl-Fredrik Nyberg Brodda's user avatar
9 votes
3 answers
581 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
  • 7,951
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 ...
Carl-Fredrik Nyberg Brodda's user avatar
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 ...
frafour's user avatar
  • 435
7 votes
1 answer
207 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
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), ...
Carl-Fredrik Nyberg Brodda's user avatar
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 ...
frafour's user avatar
  • 435
5 votes
0 answers
196 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
  • 1,267
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 ...
Francesco Polizzi's user avatar
4 votes
1 answer
130 views

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
624 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
229 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
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. ...
Matt Zaremsky's user avatar
4 votes
2 answers
222 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
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{...
Francesco Polizzi's user avatar
3 votes
1 answer
240 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
125 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
385 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