Algebra/Topology seminar

Speaker: Robert Kucharczyk

Title: On congruence subgroups of Fuchsian groups

Abstract: "Subgroups of finite index in SL(2,Z) have been intensively studied since the 19th century. These can be divided into congruence and non-congruence subgroups. The former feature prominently in the theory of elliptic curves and have many remarkable special properties, but are much rarer. A similar distinction can be made for general arithmetic subgroups of SL(2,R).

In my talk I will argue that congruence subgroups can also be defined in a natural way for some non-arithmetic lattices in SL(2,R) appearing in algebraic geometry, and present some results suggesting that this generalised notion again produces a sharp dichotomy for these groups. These include a rigidity theorem for lattices in SL(2,R) that serves as a replacement for Mostow's rigidity theorem (which is false for SL(2,R)) and makes use of this generalised notion of congruence subgroups, hence acquiring an adelic flavour."