Klara Stokes: Pentagonal geometries - an alternative way to generalise the pentagon

Partial linear spaces are combinatorial arrangements of points and 
lines. A generalised polygon is a partial linear space such that its 
bipartite incidence graph has girth twice its diameter, just like for 
ordinary polygons - the incidence graph of the ordinary n-gon is the 
cyclic graph on 2n vertices. By the Feit-Higman Theorem, a 
(non-degenerate) generalised polygon has diameter n either 3, 4, 6, or 
8. In particular there are no generalised pentagons or generalised 
heptagons. In this talk I will describe an alternative way of 
generalising the pentagon: the pentagonal geometries. A pentagonal 
geometry is a partial linear space in which for all points p, the points 
not collinear with the point p, form a line. I will talk about available 
constructions, some non-existence results, a connection with 
distance-regular graphs and embeddings on Riemann surfaces.