In the 1960s Thoma developed a theory of characters which generalizes the classical Fourier/Pontryagin theory of abelian groups, and at the same time Frobenius' theory on finite (and compact) groups.

After presenting the general theory, I will focus on arithmetic groups, or similarly, lattices in (semi)simple Lie groups, and tell about my work with Levit and Slutsky regarding the geometry/topology of the space of characters of such groups. Our main result is that for lattices in higher rank simple Lie groups (e.g for the group SL3(Z)), any sequence of distinct characters must converge pointwise to the dirac character at the identity.  This implies character bounds of finite groups of Lie type (e.g SL3(Fp)).