Questions tagged [combinatorial-group-theory]

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12
votes
1answer
270 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
1answer
225 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
1answer
128 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. $\...
6
votes
0answers
66 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
1answer
73 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
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0answers
114 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 ...
2
votes
2answers
243 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
1answer
107 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
1answer
582 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
1answer
179 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 ...
16
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1answer
424 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
2answers
169 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 ...
7
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0answers
382 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
1answer
193 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 ...
0
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0answers
55 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(\...
9
votes
2answers
316 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 ...
3
votes
1answer
105 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 ...
2
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0answers
99 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 ...
5
votes
0answers
143 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 $F$. ...
12
votes
1answer
624 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 ...
4
votes
1answer
152 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? ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
15
votes
2answers
703 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 ...
8
votes
1answer
187 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 ...
6
votes
0answers
328 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 ...
8
votes
0answers
239 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'...
6
votes
0answers
453 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 $m$ ...
5
votes
3answers
186 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 ...
12
votes
1answer
746 views

Is there a name of semidirect product of a group with its automorphism group?

Consider the construction $G \rtimes \text{Aut}(G)$. Here $ G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action. 1) Is there any name ...
3
votes
1answer
252 views

When is the semidirect product of $(Z/pZ)^n$ and $(Z/qZ)^2$ generated by two elements?

I asked a very similar question here and got a wonderful answer. But now I need to change the question slightly (this is the last question like this, I promise). I would like to characterize when $(\...
9
votes
1answer
544 views

When is the semidirect product of an elementary abelian group and a cyclic group generated by two elements?

I am trying to characterize when a semi-direct product of the form $(Z/pZ)^n \rtimes (Z/qZ)$ is isomorphic to a group generated by two elements. Here $p$ and $q$ are distinct odd primes. I would be ...
11
votes
0answers
324 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 ...
2
votes
1answer
289 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 ...
21
votes
1answer
630 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 ...
6
votes
1answer
401 views

Minimum number of operations necessary to arrive at any configuration

Let $k \geq 2$ and $N_1, N_2, ..., N_k$ be positive integers. Let $S=\{(a_1,a_2,...,a_k) \in \mathbb{Z}^k:1 \leq a_i \leq N_i\}$ and $A=\{1,2,...,\prod_{i=1}^{k} N_{i}\}$. Given a bijective map $f:...
10
votes
1answer
145 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 ...
6
votes
1answer
358 views

An algorithm determining whether two subgroups of a finitely generated free group are automorphic

In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two ...
10
votes
0answers
167 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^{-...
12
votes
1answer
312 views

“Bisecting” a free subgroup with respect to word length

My broad question is regarding the lengths of (reduced) words in a subgroup of a free group. As motivation, consider the free group $Gp(S)$ where $|S|=n$, that is, a free group of rank $n$. Let $S=\{...
3
votes
1answer
106 views

Maximal power in a sequence of iterated commutators in the rank two free group

I have the following problem: in the free group $F_2=\langle a,b\rangle$, we define the sequence $\begin{cases} w_0=a, \\ w_1=b, \\ w_{n+2}=[w_{n+1},w_{n}] & \text{for }n\ge 0. \end{cases}$ So $...
2
votes
0answers
129 views

Concentration of Reduced words

This might be a rather broad question, and I'll be satisfied with some intuition and pointers to relevant literature. However, I'll certainly fill in more context and details based on any feedback. ...
2
votes
2answers
415 views

Subgroup of a free group that is characteristic but not totally characteristic

Looking for a counter example (if it exists) and a reference for further reading. Can there be a subgroup of finite index in a finitely generated free group that is characteristic but not totally ...
2
votes
0answers
90 views

Change of generators and shortest product in groups

Let $G$ be a finitely generated group. For a set of generators $B$ of $G$, $\ell_B(x)$ is the length of the smallest sequence of elements(and inverse of the elements) in $B$, such that the product ...
3
votes
2answers
237 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 ...
4
votes
0answers
114 views

Order problem in nilpotent groups

Let $G$ be a f.g. nilpotent group. I wanted to know if the order problem (given $g \in G$, deciding if there exists $n$ s.t. $g^n=e$) is decidable in $G$? In such a group, the word problem is ...
8
votes
1answer
240 views

Need a good name for an algorithmic problem in groups that generalizes the conjugacy problem

I am looking for a good name for the following problem: Given elements $g_1,\dotsc,g_n$ in a (finitely generated) group $G$, determine if the product of their conjugacy classes $g_1^G\dotsb g_n^G$ ...
2
votes
0answers
190 views

When can we establish an isomorphism between two not-finitely presented groups?

Assume that finitely generated groups $G$ and $H$, are not finitely presented. Fix a generating set $\mathfrak g:=\{g_1,\dotsc,g_n\}$ of $G$. Let $\mathfrak R:=\{R_1,R_2,\dots\}$ be the set of all ...
2
votes
1answer
154 views

Could the number of defining relators of a finitely presented group increase

Do there exist finitely generated groups $G$ and $H$ with following properies: $G=\langle g_1,\dotsc,g_n\rangle$ is not finitely presented, Let $R:=\{r_1,r_2,\dots\}$ be the set of its defining ...
4
votes
2answers
416 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 ...