Commuting tuples in reductive groups and their maximal compact subgroups

Fundamental groups of character varieties: surfaces and tori

“…By the main results in [16] and [29], Hom 0 (Z k , DG) is simply connected. Note here that DG has the form F k × H 1 × · · · × H n , where F is either R or C and the H i are simply connected, simple Lie groups (see [25,Section 2], for instance).…”

“…In the case when is free Abelian, the main results in [16] and [29] imply that the k-th factor inclusions induce an isomorphism π 1 (DG) r ∼ = −→ π 1 (Hom 0 ( , DG)). …”

The topology of Baumslag–Solitar representations

“…Nevertheless, recent success in producing such retractions has been achieved using a mixture of topological and algebraic tools for classical families of finitely generated groups. Notable positive results include the case of free-abelian groups by Pettet-Souto [9], torsion-free expanding nilpotent groups by Silberman-Souto [13] and general nilpotent groups by Bergeron [2].…”

On the fundamental group of $${{\rm Hom}({\mathbb Z}^k,G)}$$

“…However, we would like to mention that the two last authors of this note have proved in [8] that if G is the group of complex points of a connected reductive algebraic group and K is a maximal compact subgroup, then the inclusion of Hom(Z k , K ) into Hom(Z k , G) is a homotopy equivalence. Since also G and K are homotopy equivalent, we deduce from Theorem 1.1: Corollary 1.2 Let G be the group of complex points of a connected reductive algebraic group.…”