An explicit upper bound for the Helfgott delta in SL(2, p)

Helfgott proved that there exists a δ > 0 such that if S is a symmetric generating subset of SL(2, p)containing 1 then either S^3=SL(2, p)or |S^3| ≥|S|^1+ δ. It is known that δ ≥ 1/3024. Here we show that δ ≤ (log_2 (7) −1)/6 ≈ 0.3012and we present evidence suggesting that this might be the true value of δ.This is the in press, corrected proof version of the article published by Elsevier in the Journal of Algebra online here: http://www.sciencedirect.com/science/article/pii/S0021869314004888. It will be replaced by the final published version when it becomes available

A Study of the Free Product Group with a Generator of Order 2 and a Generator of Order 3

The modular group PSL(2,Z) is well known to be the group of linear fractional transformations of the upper half of the complex plane. It is isomorphic to the quotient SL(2,Z)/K of the special linear group by its center K = {-I, I}. It acts on the hyperbolic plane as a discrete subgroup of PSL(2,R). Our work concerns the identification of quotients of PSL(2,Z) by the normal subgroups N(g) generated by a single element g. We will demonstrate great variation in the quotient types depending on the parity of the length of generating word, and apply these differences through evaluating the Cayley diagrams of these quotient groups. We will also develop an understanding of these N(g) by the action of PSL(2, Z) and its quotient groups on H, the upper half-plane model of hyperbolic space

A comment on the bianchi groups

In this paper, we aim to discuss several the basic arithmetic structure of Bianchi groups. In particularly, we study fundamental domain and directed orbital graphs for the group PSL(2;O_1)

Vertex-transitive triangulations of compact orientable 2-manifolds

AbstractIn this note we construct two infinite families of vertex-transitive triangulations of compact orientable 2-manifolds. Included in these families are two of the best known “classical” examples, viz., the triangulation of the genus 3 surface admitting the group PSL(2, 7) and the triangulation of the genus 7 surface admitting SL(2, 8)

Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

The Grothendieck--Teichmüller group of~PSL(2,q)

International audienceWe show that the Grothendieck-Teichmüller group of PSL(2, q), or more precisely the group GT 1 (PSL(2, q)) as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, when q is even, we show that it is trivial. We explain how it follows that the moduli field of any "dessin d'enfant" whose monodromy group is PSL(2, q) has derived length ≤ 3