Title. The role of twisted wreath products in the finite congruence lattice problem
Speaker. Péter Pal Pálfy
Institution. Rényi Institute, Hungarian Academy of Sciences, Budapest
Abstract. On the poster of the conference we see Bill, JB and Ralph looking at a blackboard showing a particular 7-element lattice. This is the smallest lattice that is not known to be representable as the congruence lattice of a finite algebra (William DeMeo’s Thesis, UH, 2012).
Although I am not able to solve the representation problem for this lattice, I will show how to reduce the problem to representing finite lattices as intervals in subgroup lattices of finite groups of two particular types: (1) almost simple groups; (2) twisted wreath products with peculiar properties. This is based on works of Baddeley, Lucchini, Börner, Aschbacher, and Shareshian. Twisted wreath products were introduced by B. H. Neumann in 1963. I will indicate how naturally this group theoretic construction arises and what relevance it has for the finite congruence lattice representation problem.