9th meeting: Groups & Combinatorics

Birkbeck, University of London, Wednesday December 14th 2016

All talks will be held in room B04 of the main Malet Street building. Directions available on request from b.fairbairn [at ] bbk.ac.uk if needed.

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).

13:30-14:30 Atiqa Sheikh (Imperial) Orbital diameters of the alternating and projective special linear groups

In this talk we explore a particular family of edge-transitive graphs called orbital graphs. The orbital diameter of a primitive group G acting on a finite set Ω is defined to be the maximum of the diameters of the orbital graphs of G. We will give an overview of the results obtained on the orbital diameters of groups whose socle is an alternating group or a projective special linear group of dimension 2. The results concern boundedness of the orbital diameters and classifying orbital graphs of small diameters.

14:30-15:30 Jozef Širáň (OU) Regular maps arising from twisted linear groups

A map a cellular embedding of a graph on a surface. A flag of a map is a triple consisting of a vertex, an edge and a face side that are mutually incident. An automorphism of a map is a permutation of its flags such that two flags sharing a vertex, edge or a face are sent to a pair of flags sharing the same type of elements. The collection of all such mappings under composition forms the automorphism group of a map, acting semi-regularly on the flag set. If this action is regular we say that the map is regular as well. Regular maps may be viewed as having the ‘highest level of symmetry’ among all maps. A slightly weaker concept is that of an orientably-regular map, where only orientable surfaces and orientation-preserving automorphisms are considered.

Regular maps embody three varieties of objects: a graph, a surface on which the graph is embedded, and a group isomorphic to the automorphism group of the embedding. Classification of regular and orientably-regular maps, which is a central problem in the theory of symmetric maps, is therefore usually attempted for a particular collection of underlying graphs, or carrier surfaces, or automorphism groups.

In the talk we will focus on classification of regular and orientably-regular maps with a given (isomorphism type of) automorphism group. We will give a survey of available results; as it turns out, two-dimensional linear groups
PSL(2,q) and PGL2,q) appear to be best understood from this point of view. Interestingly, the PGL(2,q) groups are one of the two infinite families of finite sharply 3-transitive permutation groups. Their ‘counterpart’, however, i.e., the ‘other’ such infinite family consisting of the so-called twisted linear groups, have been somewhat neglected in the literature. We will also present new results on enumeration of regular and orientably-regular maps with twisted linear automorphism groups and discuss open questions in this area of research.

15:-16:00 Break

16:00-17:00 William Norledge (Newcastle) Covering Theory of Combinatorial Buildings

Buildings are certain simply connected and highly symmetrical polyhedral complexes, which have a combinatorial description as a “Weyl-metric space”. Towards a greater understanding of locally compact groups, the groups which act on buildings are studied in a hope to extend the work of Bruhat-Tits on algebraic groups over non-Archimedean local fields acting on Euclidean buildings.
Covering theory of complexes of groups has been used to construct actions of groups on buildings. During the talk, we’ll introduce covering theory of buildings which doesn’t resort to complexes of groups, but rather deals directly with the Weyl-metric. We’ll use this covering theory to construct actions of amalgams of surface groups on hyperbolic buildings.

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