Questions tagged [combinatorial-group-theory]
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168 questions
1 vote
0 answers
61 views
Presentation of the mixed Braid group
This question concerns the mixed Braid groups $B_{n,P}$. Suppose $P$ is a partition of $\{1, \ldots, n\}$. Consider the subgroup $S_{n,P}$ of the symmetric group $S_n$ consisting of the permutations ...
- 737
4 votes
1 answer
236 views
RT-structures in finite groups
During my research in Algebraic Geometry, I was led to the following problem in Combinatorial Group Theory, strictly related to finite quotients of pure surface braid groups. Let $G$ be a finite group....
- 68.3k
0 votes
0 answers
70 views
Are the presentations of free profinite group as an one-relator group essentially unique?
Let $F$ be the free profinite group of rank $r$. Let $\tilde{F}$ be the free profinite group of rank $r + 1$, and $C$ be a subgroup of $\tilde{F}$ which is, of course non-canonically, isomorphic to $\...
- 103
3 votes
1 answer
210 views
Commuting elements in a right-angled Artin group
Given a right-angled Artin group (RAAG) $A_\Gamma$ it is well-known that two generators $u$ and $v$ commute if and only if $u$ and $v$ are connected with an edge in $\Gamma$. But, is there are general ...
- 1,009
9 votes
4 answers
565 views
How many translates of the singular‐matrix hypersurface are needed to cover $M_n(\mathbb{F}_2)$?
Let $n$ be a positive integer, and consider the hypersurface of singular $n\times n$ matrices over $\mathbb{F}_2$, denoted $$ \mathcal{S}_n = \{X\in M_n(\mathbb{F}_2) : \det(X)=0\}. $$ Note that \...
- 469
9 votes
1 answer
418 views
Is this cyclically pinched one-relator group fully residually free?
Consider the following one-relator group: $$G = \langle x,y,w,z \mid x^3[x,y][w,z] = 1\rangle$$ where $[a,b] = aba^{-1}b^{-1}$ denotes the commutator. It is the free product with amalgamation $$F(x,y) ...
- 515
1 vote
0 answers
114 views
Average Whitehead minimized length
Let $w$ be a word uniformly sampled from the cyclically reduced word in the free group on $r$ elements. I'm looking for the expected length of the Whitehead minimization of $w$. I don't need a precise ...
- 491
4 votes
1 answer
244 views
Variant of the coupon collector problem on free groups
I have the following variant of the birthday problem / coupon collector problem: Let $w$ be a reduced word sampled uniformly from the set of reduced words of length $n$ in the free group $F_r$. As $n$...
- 491
5 votes
0 answers
183 views
Hard word problems for natural groups
Are there known examples of naturally-occurring groups where the word problem is algorithmically solvable but not easily? I ask because I'm looking at word problems of some groups of interest to me, ...
- 6,934
7 votes
0 answers
164 views
Word problem for finite colimit of finite groups
Let $I$ be a finite category and $F \colon I \rightarrow \mathrm{Grp}$ be a functor into the category of groups, such that $F(c)$ is a finite group for every $c \in Obj(I)$. Question. Does $\mathrm{...
- 541
2 votes
1 answer
227 views
Growth rate of a free group with slow growing sequence of forbidden subwords
Let $F_n$ be the free group on $n$ generators where $n \geq 2$. Let's say I have a sequence of words $W=\{w_i\}_{i=0}^\infty$ such that: $w_i \sqsubseteq w_j$ ($w_i$ is a subword of $w_j$) iff $i=j$. ...
- 491
5 votes
2 answers
499 views
Centre of group with deficiency at least two (Progress on Murasugi's conjecture)
In 1965, Murasugi [1] conjectured that any finitely presented group with deficiency at least two has trivial centre. The year before, he had proved it true for one-relator groups, and in [1] he proved ...
8 votes
1 answer
473 views
Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
4 votes
1 answer
364 views
Are (group theoretic) Markov properties on groups with decidable word problems, decidable?
(Link to SE duplicate: https://math.stackexchange.com/questions/4959071/are-group-theoretic-markov-properties-on-groups-with-decidable-word-problems) The Adian-Rabin theorem says that if a property of ...
- 191
7 votes
1 answer
582 views
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
210 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 ...
- 1,774
5 votes
1 answer
351 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=\...
- 53
2 votes
1 answer
399 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 ...
- 4,494
2 votes
0 answers
89 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 ...
- 1,063
2 votes
0 answers
192 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 $n\in \mathbb{Z}$, $...
- 1,063
3 votes
0 answers
466 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: ...
- 403
18 votes
1 answer
798 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
360 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 ...
- 3,143
16 votes
2 answers
1k 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 ...
- 1,063
7 votes
1 answer
617 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 ...
- 403
12 votes
1 answer
354 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 ...
- 9,731
1 vote
0 answers
101 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 ...
- 435
2 votes
1 answer
297 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 ...
- 1,009
10 votes
3 answers
769 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 ...
4 votes
2 answers
297 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 $H=\langle X\cup X'\mid R\cup R'\rangle$, ...
- 1,774
5 votes
0 answers
230 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 ...
- 179
3 votes
0 answers
162 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_{\...
- 1,063
4 votes
0 answers
273 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$...
- 1,063
6 votes
2 answers
678 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?
user486228
3 votes
2 answers
265 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 ...
- 302
10 votes
2 answers
1k 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 ...
- 435
17 votes
3 answers
2k 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 (...
- 435
2 votes
1 answer
335 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 ...
- 2,086
5 votes
1 answer
440 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 ...
- 1,063
5 votes
0 answers
186 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\...
- 950
4 votes
1 answer
338 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 ...
- 1,063
1 vote
1 answer
332 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 ...
- 4,706
9 votes
1 answer
463 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?" ...
- 133
9 votes
1 answer
466 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 ...
- 93
7 votes
0 answers
207 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
363 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 ...
10 votes
3 answers
905 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 ...
- 8,861
12 votes
1 answer
467 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
313 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 ...
- 435
7 votes
1 answer
267 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. $\...