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 ...

I'm interested in the affine Weyl group $W$ of type $\tilde A_{n-1}$, but thequestion makes sense in any type. The structure of this group is that it sits in a split short exact sequence $1\to \...

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_{\...

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 ...

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 ...

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 $2n$ ...

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?

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 ...

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 $G$ ...

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 = ...

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 ...

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 ...

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 ...

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 ...

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 ...