Abstract

Let G be a noncompact simple real Lie group, and let Γ be an irreducible discrete subgroup of G × G, i.e. its projections to both factors are dense. For G = SL(2,R), the only known examples are lattices. We will highlight some questions and present some results showing that certain irreducible groups are subject to strong structural constraints. Along the way, we construct counterexamples to a conjecture of Margulis by exhibiting discrete subgroups of infinite covolume in higher-rank simple groups G whose associated locally symmetric spaces admit no nontrivial bounded harmonic functions. We will also discuss examples of two quasi-isometric locally symmetric spaces with finitely presented fundamental groups, where one admits nontrivial bounded harmonic functions while the other does not, highlighting ergodic properties at infinity. This is all based on joint work in progress with Sebastian Hurtado.