Algebra Seminar: Thomas -- Platonic lattices for trivalent Platonic polygonal complexes

Anne Thomas (University of Sydney) 

Friday 24 May, 12-1pm, Place: Carslaw 375 

Title: Platonic lattices for trivalent Platonic polygonal complexes 

Abstract: A polygonal complex X is Platonic if Aut(X) acts transitively on the set of
flags (vertex, edge, face) of X.  Trivalent Platonic polygonal complexes X of
nonpositive curvature were investigated by Swiatkowski, who determined in particular
when the locally compact group Aut(X) is nondiscrete.  For these X, we use triangles of
finite groups to investigate the existence of Platonic lattices in Aut(X), where a
Platonic lattice is a discrete subgroup of Aut(X) which acts flag-transitively.  Our
constructions include the first examples of lattices in nondiscrete Aut(X) where X is
neither a building nor a Davis complex.  By results of Haglund, Agol and Haglund-Wise,
the Platonic lattices we obtain are linear groups.  This is joint work with Inna
Capdeboscq and Michael Giudici.