This thesis studies the geometry of representation varieties and character stacks. These are spaces parametrizing the representations of a finitely generated group, typically the fundamental group... Show moreThis thesis studies the geometry of representation varieties and character stacks. These are spaces parametrizing the representations of a finitely generated group, typically the fundamental group of a compact manifold, into an algebraic group G. We provide a number of methods to study these spaces through their invariants, focussing on the invariant that takes values in the Grothendieck ring of varieties. We show how both the arithmetic method, which studies the character stacks of compact orientable surfaces through counting points over finite fields, and the geometric method, which studies these character stacks using stratifications, can be expressed in terms of topological quantum field theories. We compute explicitly the invariants in specific cases, such as for G = SL2 and for G equal to the groups of upper triangular matrices. Motivated by these applications, we develop a number of new computational tools. Finally, we study the representation varieties and character stacks of the free groups and of the free abelian groups, of finite rank. We introduce a notion of motivic representation stability for stability in the Grothendieck ring of varieties, and show that these character stacks stabilize in this sense for G equal to the general linear groups. Show less
