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11
votes
2answers
257 views

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 ...
7
votes
2answers
183 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 ...
4
votes
2answers
452 views

Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $GL_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More precisely, the ...
0
votes
1answer
225 views

Ordered groups - examples

Let $G=BS(m,n)$ denote the Baumslag–Solitar groups defined by the presentation $\langle a,b: b^m a=a b^n\rangle$. We assume that G is non-abelian. Question: Find $m,n$ such that $G$ is an ...
9
votes
1answer
467 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 ...
3
votes
1answer
250 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 ...
1
vote
1answer
76 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 ...
3
votes
0answers
193 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 ...
0
votes
0answers
206 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 ...
0
votes
1answer
84 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 ...
5
votes
1answer
178 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 ...
7
votes
3answers
608 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.) ...
2
votes
1answer
233 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 ...
3
votes
1answer
184 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 ...
2
votes
2answers
393 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 $H$ ?
6
votes
1answer
200 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 ...
4
votes
3answers
413 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 ...
0
votes
1answer
163 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 ...
2
votes
0answers
144 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 $D_n$ ...
22
votes
1answer
2k 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 ...
4
votes
3answers
564 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 ...
2
votes
2answers
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.
3
votes
2answers
315 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 ...
24
votes
15answers
6k views

What's so great about blackboards? [closed]

Many mathematicians seem to think that the only way to give a mathematics talk is by using chalk on a blackboard. To some, even using a whiteboard is heresy. And we Don't Talk About Computers. I'd ...
0
votes
2answers
412 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 ...
8
votes
3answers
469 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
4answers
4k 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!) ...