Non-arithmetic lattices Seminar

A lattice in a semi-simple Lie group G is a discrete subgroup whose quotient has finite Haar measure. It acts on the associated symmetric space a a discrete group of isometries with finite covolume. An arithmetic subgroup of a linear algebraic group is a subgroup that is discrete because the integers are discrete in the real numbers. All lattices are arithmetic except when G is SO(n,1) or SU(n,1). The cases of SO(n,1 and SU(1,1) are quite well understood. There are 9 examples (up to commensurability) of non-arithmetic lattices in SU(2,1) and there is a single example in SU(3,1). These are all due to Deligne and Mostow in 1986. For n at least 4 the problem is open. In joint work with Deraux and Paupert we have found 10 more examples of non-arithmetic lattices in SU(2,1), the first to be found since 1986. I will give a gentle introduction to the problem and outline how we found our new examples.