Questions tagged [combinatorial-group-theory]

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2 votes
0 answers
35 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
239 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=\...
3 votes
0 answers
447 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 ...
2 votes
1 answer
321 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 ...
2 votes
0 answers
57 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 ...
2 votes
0 answers
146 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 $n\in \mathbb{Z}$, $...
3 votes
0 answers
206 views

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
724 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
157 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 ...
16 votes
1 answer
791 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 ...
7 votes
1 answer
549 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
180 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 ...
10 votes
2 answers
775 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 ...
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 ...
2 votes
1 answer
209 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
459 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 ...
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 $G$ ...
4 votes
2 answers
207 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
185 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 ...
3 votes
0 answers
126 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_{\...
4 votes
0 answers
193 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 $\phi$...
6 votes
2 answers
523 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
182 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 ...
10 votes
2 answers
747 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 ...
14 votes
1 answer
729 views

Finite-dimensional version of the word problem for groups

The (uniform) word problem for groups can be stated in several equivalent ways: Word Problem for Groups (WP) Instance: A finite presentation of a group G and an element w of G as a product of ...
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 (...
2 votes
1 answer
197 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 ...
4 votes
1 answer
276 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 ...
5 votes
1 answer
369 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
128 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\...
11 votes
2 answers
745 views

Is every virtual knot group an HNN extension?

A basic fact in knot theory is that a knot group $\pi(K)$ is an HNN extension of $\pi(F)$, the fundamental group of a Seifert surface complement. A nice discussion of this may be found in Chapter 11 ...
1 vote
1 answer
230 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
330 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
353 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
176 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{...
7 votes
0 answers
259 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
542 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 ...
9 votes
2 answers
698 views

Embedding of Baumslag-Solitar group into a certain group

Let $G$ be a group generated by $a_0, a_1, a_2$ with relations: $a_0 a_1 a_0^{-1}=a_1^4$ $a_1 a_2 a_1^{-1}=a_2^4$ $a_2 a_0 a_2^{-1}=a_0^4$ I am wondering if $BS(1,4)=\langle a,b:bab^{-1}=a^4\...
12 votes
1 answer
378 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
3 answers
785 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 ...
9 votes
1 answer
274 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
204 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. $\...
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 ...
8 votes
0 answers
110 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
172 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
192 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 ...
4 votes
1 answer
129 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 ...
2 votes
2 answers
285 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 ...
2 votes
1 answer
623 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 ...
2 votes
1 answer
227 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 ...