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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 ...
Chitrabhanu's user avatar
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1 answer
240 views

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....
Francesco Polizzi's user avatar
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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 $\...
Alice's user avatar
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3 votes
1 answer
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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 ...
Marcos's user avatar
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4 answers
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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 \...
Luftbahnfahrer's user avatar
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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) ...
HASouza's user avatar
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0 answers
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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 ...
Atsma Neym's user avatar
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1 answer
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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 $n$...
Atsma Neym's user avatar
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0 answers
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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, ...
Ville Salo's user avatar
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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{...
user68822's user avatar
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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 $i=j$. ...
Atsma Neym's user avatar
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2 answers
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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 ...
Carl-Fredrik Nyberg Brodda's user avatar
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8 votes
1 answer
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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 \...
Carl-Fredrik Nyberg Brodda's user avatar
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4 votes
1 answer
370 views

(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 ...
Perry Bleiberg's user avatar
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1 answer
585 views

We consider Artin-Tits groups of two generators $(I_2(n))$. Are these groups ordered groups?
navashree chanania's user avatar
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1 answer
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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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1 answer
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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. $$G=\...
keqiyehuopo's user avatar
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[Cross-posted from MSE] Consider the Von Dyck group $$ 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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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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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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3 votes
0 answers
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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
802 views

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
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3 votes
2 answers
362 views

$\DeclareMathOperator\rank{rank}$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
2 answers
1k views

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
620 views

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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12 votes
1 answer
356 views

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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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
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2 votes
1 answer
303 views

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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10 votes
3 answers
788 views

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
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4 votes
2 answers
298 views

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
232 views

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
162 views

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
276 views

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
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6 votes
2 answers
688 views

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?
user avatar
3 votes
2 answers
268 views

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
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10 votes
2 answers
1k views

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
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17 votes
3 answers
2k views

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
  • 457
2 votes
1 answer
342 views

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
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5 votes
1 answer
448 views

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
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5 votes
0 answers
186 views

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
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4 votes
1 answer
338 views

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
  • 1,063
1 vote
1 answer
337 views

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
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9 votes
1 answer
478 views

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
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9 votes
1 answer
469 views

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
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7 votes
0 answers
207 views

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
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7 votes
0 answers
372 views

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
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10 votes
3 answers
916 views

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
  • 8,861
12 votes
1 answer
467 views

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
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9 votes
1 answer
314 views

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
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7 votes
1 answer
275 views

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
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