Alastair Litterick

Description

(York Algebra Seminar Online)

Title: Reductive groups, representation varieties, rigidity

Abstract: In this talk we will consider the following three problems: (1) Understanding finite simple quotients of finitely-generated groups; (2) Understanding generating sets in finite groups; (3) Understanding the subgroup structure of reductive algebraic groups.
The punchline is that all three of these problems come down to studying a space Hom(F,G), where F is finitely generated and G is a reductive group.

The set Hom(F,G) is an algebraic variety, the representation variety of F in G, with a natural G-action. Using algebraic geometry and geometric invariant theory we are able to prove a ‘rigidity’ result: under natural hypotheses, the G-orbits of certain interesting homomorphisms are both closed and open in an appropriate subvariety of Hom(F,G); this is a very strong condition. As an application, if F is generated by torsion elements which multiply to 1, if G is defined over a finite field Fq, and if a certain ‘dimension bound’ holds in G, then only finitely many groups of Lie type G(qe) are quotients of F. This proves and generalises a 2010 conjecture of C. Marion on triangle groups.

Period	16 Jun 2020
Visiting from	University of Essex (United Kingdom)
Visitor degree	PhD
Degree of Recognition	Local