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 finitelygenerated 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 Gaction. Using algebraic geometry and geometric invariant theory we are able to prove a ‘rigidity’ result: under natural hypotheses, the Gorbits 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 
Related content

Activities

Department of Mathematics  Algebra Seminar Series
Activity: Participating in or organising an event › Seminar/workshop/course