Questions tagged [presentations-of-groups]

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Write an Artin group as an HNN-extension

Assume that $A_\Gamma$ is an Artin group and $\chi:A_\Gamma\to(\mathbb{Z},+)$ is a group homomorphism of the following form. $\Gamma=\Gamma_1\cup\Gamma_2$ with $\Gamma_1\cap\Gamma_2=\emptyset,A_{\...
Marcos's user avatar
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Presentation of handlebody mapping class group

I know some 'nice' infinite presentations of the mapping class group of a surface, such as Gervais' and Luo's. By 'nice' I mean that generators and relations belong to a small number of families. Is ...
Filippo Bianchi's user avatar
4 votes
1 answer
345 views

Group presentation in the category of finite group

Context: I'm trying to deal with presentations in the framework of Gonthier et al. formalization of the group theory in the proof assistant Coq. It was used to machine check the Feit-Thompson odd ...
hivert's user avatar
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3 votes
1 answer
264 views

Direct proof (or reference) that a given $p$-group is extra-special

Writing a paper on algebraic surfaces, I was led to consider the finite group $\mathsf{H}(A)$ whose presentation is the following. I start with an anti-symmetric matrix $A=(a_{ij})$ of order $2n$ ...
Francesco Polizzi's user avatar
3 votes
2 answers
181 views

Presentations of superperfect groups

Are there non-trivial superperfect groups with the property that there exists a presentation of the group where the number of generators equals the number of relations?
Sergei Nikodimov's user avatar
2 votes
1 answer
212 views

Mapping $\Delta(2,2,2)\mapsto \Delta(4,4,2)$

Looking at the images below, you recognize that the adjacency matrix of the graph $A_G$ splits up into three different colored submatrices, with $A_G=A_r+A_b+A_d$ (where $d$ is dark, damn...). It's ...
draks ...'s user avatar
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1 vote
1 answer
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What happens when you internalize outer automorphisms?

Given a finitely presented group $G = (Gen|Rel)$, we have a set of inner automorphisms $\{ \phi_a(x) = axa^{-1} | a \in G\}$. Defining the set of outer automorphisms to be those automorphisms of $G$ ...
Samuel Schlesinger's user avatar
2 votes
1 answer
525 views

A presentation for $GL(2,\mathbb{Z}/p^n \mathbb{Z})$

In 'A presentation of $PGL(2,p)$ with three defining relations' by E.F.Robertson and P.D.Williams, we can find a presentation of $PGL(2,p)$: $\langle a,b | a^2 = b^p = (a b^2 a b^r)^2 = (abab^r)^3 = ...
user48096's user avatar
  • 153
0 votes
1 answer
193 views

Must a group of defficiency > 1 be nonabelian?

Let $F$ be a free group of finite rank $n > 2$, and let $S \subseteq F$ be a subset of cardinality at most $n-2$. Denote by $S^F$ the normal subgroup of $F$ generated by $S$. Must $F/S^F$ be ...
Pablo's user avatar
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16 votes
2 answers
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Algorithms in hyperbolic groups

I'm stuck in some algorithms in hyperbolic groups, which may be rather simple. Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can ...
stephen's user avatar
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9 votes
3 answers
480 views

Modifying Dehn's algorithm to allow equal length replacements?

I'm an analyst trying to understand a certain class of finitely presented groups (one example is below) so it's quite likely this question is naive but I hope it is at least intelligible. Given a ...
Mike Jury's user avatar
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4 votes
1 answer
326 views

Quotient of Coxeter Group II

My last question on the quotients of the group $$H := \langle a, b, c \ | \ a^2, b^2, c^2, (ab)^2, (ac)^3, (bc)^7, (abc)^{19} \rangle$$ couldn't be completely answered, because the finiteness of the ...
Thomas's user avatar
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11 votes
1 answer
864 views

Why isn't $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}\rangle$ a hyperbolic surface group?

The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed torus fundamental ...
Brian Rushton's user avatar
3 votes
1 answer
372 views

Cohomological dimension of groups & number of generators

I have a torsion-free non-abelian nilpotent group $\Gamma$ of cohomological dimension $n$. Is it possible to say anything about the number of generators of $\Gamma$ in a minimal presentation? Can I ...
Earthliŋ's user avatar
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1 vote
1 answer
110 views

Do Nielsen transformations on a presentation preserve the homotopy type of the corresponding presentation complex?

Let $\mathcal{P}$ be a finite presentation of some group. When we apply some Nielsen transformations on $\mathcal{P}$, will the homotope type of the presentation complex $K_{\mathcal{P}}$ of $\mathcal{...
Li Yu's user avatar
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0 answers
438 views

The second homology of a group G and presentation complex of G

Let $G$ be a finitely presentable group. If we assume $H_2(G,Z/pZ) =0$, $p$ is a prime, then can we always find a finite presentation $\mathcal{P}$ of $G$ so that its presentation complex $K_{\mathcal{...
Li Yu's user avatar
  • 341
0 votes
0 answers
397 views

Finitely presented group and its subgroups

Suppose I have a finitely presented group $G$. By this, I mean I know explicitly what $S$ and $R$ are such that $G = \langle S \mid R \rangle$. Suppose I have a subgroup generated by a finite set of ...
expmat's user avatar
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0 votes
1 answer
142 views

Polycyclic group not of type $FP_\infty$

In finitely presented groups, the question of the existence of a projective resolution $P_i$ (with each $P_i$ finitely generated) of $\mathbb{Z}G$ is equivalent to the existence of a $K(G,1)$ which ...
ARG's user avatar
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6 votes
2 answers
974 views

Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $\operatorname{GL}_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More ...
expmat's user avatar
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6 votes
1 answer
508 views

Relations in a particular subgroup of the braid group.

I think this should be a 10 minute exercise in a decent computer algebra package - unfortunately I'm hopelessly ignorant of such things, so I'm putting it up here in the hope that someone will be kind ...
Ed Segal's user avatar
  • 460
3 votes
1 answer
359 views

Simplifying presentations of modular subgroups

I've been using the Reidemeister-Schreier process (detailed in e.g. Holt et al. - Handbook of Computational Group Theory) to find the presentations of various modular subgroups. For example, this ...
Jimeree's user avatar
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2 votes
1 answer
367 views

Presentations of infinite index subgroups

Suppose we have a finitely presented group $G$ with a concrete presentation and a subgroup $H$, generated by a finite set of elements from $G$. How to find the presentation for $H$? If $H$ has finite ...
Dan's user avatar
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8 votes
3 answers
2k views

Are congruence subgroups of the modular group finitely presented?

Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma_{0}\left(n\right)$, $\Gamma_{1}\left(n\right)$ etc.) ...
Jimeree's user avatar
  • 383
7 votes
1 answer
243 views

Asymptotics of the number of required Dehn relators in hyperbolic groups

If $G = \langle X | R \rangle$ is a $\delta$-hyperbolic group presentation, then Dehn's algorithm provides a linear time solution to the word problem, but the linear constant is horribly exponential ...
Jeff Burdges's user avatar
7 votes
3 answers
1k views

Results in the Presentation of Finite Groups

I've been looking at combinatorial group theory, but all the results seem to be about infinite groups. Are there any important results about the presentations finite groups specifically (or are useful ...
user20886's user avatar
  • 483
0 votes
1 answer
181 views

graph of the size of a complex function [closed]

Hi Here there are two graphs for two functions from $R^2\mapsto R$. Is there similar graph for the absolute value of a complex variable function $f:C\mapsto C$ that has the same point (like saddle ...
asd's user avatar
  • 163
3 votes
0 answers
189 views

Universal polygraphic factorization of strict ω-categories relative to a cobase

Recall from 1 that a cofibration of strict ω-categories is a retract of relative $I$-cell complexes, where $I$ denotes the set of boundary inclusions $\partial D^n \hookrightarrow D^n$, where $D_n$ ...
Harry Gindi's user avatar
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9 votes
3 answers
1k views

Presentations of PSL(2, Z/p^n)

As is well known, the group $PSL(2,\mathbb Z)$ is isomorphic to the free product $C_2 \ast C_3$ of cyclic groups of order $2$ and $3$. Call the generators of the cyclic groups $S$ and $T$. Problem: ...
Łukasz Grabowski's user avatar
28 votes
1 answer
4k views

A Presentation for Rubik's cube group?

Let $G$ be Rubik's cube group. It is generated by the rotations by 90 degrees $L,R,D,U,F,B$ (left, right, down, up, front, behind), but what relations beyond $L^4=R^4=...=B^4=1$ do they satisfy? Thus ...
Martin Brandenburg's user avatar
3 votes
2 answers
1k views

Advice on Giving a Talk [closed]

What advice do you have for giving a talk on a mathematical research paper to people in other fields in science (not physics nor astronomy) but without lot of math background? Thanks.
4 votes
2 answers
485 views

Is there any way to check whether a group is residually solvable?

For a given group presentation of a group(finitely presented), I want to check whether it is residually solvable or not. Is there any good way to do it? Actually, I'm curious whether the finitely ...
hopflink's user avatar
  • 537
4 votes
3 answers
755 views

About the proof of Wajnryb's finite presentation of Mod(S)

I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't ...
hopflink's user avatar
  • 537
4 votes
2 answers
506 views

Presentation for an infinite index subgroup of the braid group

If $H$ is an infinite index subgroup of the braid group $\mathcal{B}_n$, is there a way to find a presentation for $H$ ?
Jim B's user avatar
  • 111
12 votes
3 answers
2k views

Database of finite presentations of used groups

Do You know any kind of database of presentations of groups? It may be on-line or off-line in form of tables, ideally case would be integrated in some Computer Algebra System. I am interested the ...
7 votes
4 answers
5k views

Beamer hints and tips [closed]

I deleted a rant from this question because I felt it detracted from the given answer to the specific question. However, beamer is the "new kid on the block" in terms of giving talks (not that new!) ...
0 votes
2 answers
432 views

How to present overlap of related sets [closed]

I have extracted URL links from a number of webpages and many of the webpages contain the same set of links (or subsets) as other webpages. I have ~1000 webpages and ~10 links per webpage. What is an ...
hoju's user avatar
  • 177