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Branch Groups  [PDF]
Laurent Bartholdi,Rostislav I. Grigorchuk,Zoran Sunik
Mathematics , 2005,
Abstract: This is a long introduction to the theory of "branch groups": groups acting on rooted trees which exhibit some self-similarity features in their lattice of subgroups.
Fundamental groups and presentations of algebras  [PDF]
Juan Carlos Bustamante,Diane Castonguay
Mathematics , 2004,
Abstract: In this note, we investigate how different fundamental groups of presentations of a fixed algebra $A$ can be. For finitely many finitely presented groups $G_i$, we construct an algebra $A$ such that all $G_i$ appear as fundamental groups of presentations of $A$.
New Presentations of Thompson's Groups and Applications  [PDF]
Uffe Haagerup,Gabriel Picioroaga
Mathematics , 2008,
Abstract: We find new presentations for the Thompson's groups $F$, the derived group $F^{'}$ and the intermediate group $D$. These presentations have a common ground in that their relators are the same and only the generating sets differ. As an application of these presentations we extract the following consequences: the cost of the group $F^{'}$ is 1 hence the cost cannot decide the (non)amenability question of $F$; the $II_1$ factor $L(F^{'})$ is inner asymptotically abelian and the reduced $C^*$-algebra of $F$ is not residually finite dimensional.
Symmetric Presentations of Coxeter Groups  [PDF]
Ben Fairbairn
Mathematics , 2011,
Abstract: We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is the Coxeter groups of types An, Dn and En, and show that these are naturally arrived at purely through consideration of certain natural actions of symmetric groups. We go on to use these techniques to provide explicit representations of these groups.
New presentations of surface braid groups  [PDF]
Juan Gonzalez-Meneses
Mathematics , 1999,
Abstract: In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.
Presentations of Graph Braid Groups  [PDF]
Daniel Farley,Lucas Sabalka
Mathematics , 2009, DOI: 10.1515/FORM.2011.086
Abstract: Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory to compute presentations of all graph braid groups, for all finite connected graphs G and all natural numbers n.
Indicable Groups and Endomorphic Presentations  [PDF]
Mustafa Gokhan Benli
Mathematics , 2011,
Abstract: In this note we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if $H$ is a finitely generated normal subgroup of a finitely presented group $G$ with $G/H$ cyclic, then $H$ has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product $H \rtimes \field{Z}$ where $H$ has finite ascending endomorphic presentation.
Refined solvable presentations for polycyclic groups  [PDF]
René Hartung,Gunnar Traustason
Mathematics , 2011,
Abstract: We describe a new type of polycyclic presentations, that we will call refined solvable presentations, for polycyclic groups. These presentations are obtained by refining a series of normal subgroups with abelian sections. These presentations can be described effectively by presentation maps which yield the basis data structure to define a polycyclic group in computer-algebra-systems like {\scshape Gap} or {\scshape Magma}. We study refined solvable presentations and, in particular, we obtain consistency criteria for them. This consistency implementation demonstrates that it is often faster than the existing methods for polycyclic groups.
Explicit presentations for exceptional braid groups  [PDF]
David Bessis,Jean Michel
Mathematics , 2003,
Abstract: We give presentations for the braid groups associated with the complex reflection groups $G_{24}$ and $G_{27}$. For the cases of $G_{29}$, $G_{31}$, $G_{33}$ and $G_{34}$, we give (strongly supported) conjectures. These presentations were obtained with VKCURVE, a GAP package implementing Van Kampen's method.
Presentations for quaternionic $S$-unit groups  [PDF]
Ted Chinburg,Holley Friedlander,Sean Howe,Michiel Kosters,Bhairav Singh,Matthew Stover,Ying Zhang,Paul Ziegler
Mathematics , 2014, DOI: 10.1080/10586458.2014.968269
Abstract: The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of Bruhat--Tits trees. We also include some discussion and experimentation related to the congruence subgroup problem, which is open when $S$ contains at least two odd primes. In the appendix, we provide code that allows the reader to compute presentations for an arbitrary finite set $S$.
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