The presentations-of-groups tag has no usage guidance.

**27**

votes

**15**answers

8k 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 ...

**23**

votes

**1**answer

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 ...

**12**

votes

**2**answers

449 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 ...

**11**

votes

**3**answers

637 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 ...

**10**

votes

**1**answer

610 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 ...

**7**

votes

**4**answers

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!) ...

**7**

votes

**2**answers

228 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 ...

**7**

votes

**1**answer

217 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 ...

**6**

votes

**3**answers

773 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.) ...

**5**

votes

**3**answers

593 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 ...

**5**

votes

**1**answer

222 views

### Is there a one relator group with property (T)?

Is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?

**5**

votes

**1**answer

251 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 ...

**4**

votes

**3**answers

626 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 ...

**4**

votes

**2**answers

337 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 ...

**4**

votes

**2**answers

554 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 ...

**3**

votes

**1**answer

284 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 ...

**3**

votes

**2**answers

435 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 $H$ ?

**3**

votes

**1**answer

233 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 ...

**3**

votes

**0**answers

266 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 ...

**2**

votes

**1**answer

257 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 ...

**2**

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.

**2**

votes

**1**answer

248 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 = ...

**2**

votes

**0**answers

153 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 $D_n$ ...

**1**

vote

**1**answer

81 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 ...

**0**

votes

**1**answer

162 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 ...

**0**

votes

**1**answer

95 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 ...

**0**

votes

**2**answers

415 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 ...

**0**

votes

**1**answer

247 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 ...

**0**

votes

**1**answer

169 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 ...

**0**

votes

**0**answers

282 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 ...