Questions tagged [combinatorial-group-theory]
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158 questions
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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
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1
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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 \...
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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 ...
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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?
3
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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 ...
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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.
$$G=\...
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Show me that I have not simplified the proof of the Adian-Rabin theorem
Let $G$ be a group with presentation $\langle x_1,x_2...,x_m|R \rangle$ and let $G'=G \ast \langle y_0 \rangle$.
Now define $y_i=y_{0}x_i$. Notice that $G'=\langle y_0,y_1,...y_m\mid R'\rangle$ for ...
- 191
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1
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Proving certain triangle groups are infinite
[Cross-posted from MSE]
Consider the Von Dyck group
$$ 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 ...
- 4,425
2
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0
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60
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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 ...
- 823
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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}$, $...
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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:
...
- 379
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751
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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 ...
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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 ...
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850
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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 ...
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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 ...
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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 ...
- 9,501
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0
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75
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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 ...
- 383
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239
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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 ...
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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 ...
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2
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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$, ...
- 1,338
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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 ...
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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 $\phi$...
- 823
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572
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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?
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197
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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
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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 ...
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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 (...
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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 ...
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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 ...
- 823
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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\...
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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 ...
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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 ...
- 4,432
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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?"
...
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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 ...
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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{...
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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 ...
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3
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666
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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 ...
- 8,401
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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 ...
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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 ...
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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. $\...
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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), ...
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1
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194
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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 ...
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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 ...
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309
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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 ...
- 66.3k
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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 ...
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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 ...
- 66.3k
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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 ...
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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.
...
- 3,730
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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 ...
- 66.3k
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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{...
- 66.3k