Cosimplicial groups and spaces of homomorphisms

Abstract

Let $G$ be a real linear algebraic group and $L$ a finitely generated cosimplicial group. We prove that the space of homomorphisms $Hom\left(L_n,G\right)$ has a homotopy stable decomposition for each $n\ge 1$. When $G$ is a compact Lie group, we show that the decomposition is $G$–equivariant with respect to the induced action of conjugation by elements of $G$. In particular, under these hypotheses on $G$, we obtain stable decompositions for $Hom\left(F_n∕{\Gamma }_n^q,G\right)$ and $Rep\left(F_n∕{\Gamma }_n^q,G\right)$, respectively, where $F_n∕{\Gamma }_n^q$ are the finitely generated free nilpotent groups of nilpotency class $q-1$.

The spaces $Hom\left(L_n,G\right)$ assemble into a simplicial space $Hom\left(L,G\right)$. When $G=U$ we show that its geometric realization $B\left(L,U\right)$, has a nonunital $E_{\infty }$–ring space structure whenever $Hom\left(L_0,U\left(m\right)\right)$ is path connected for all $m\ge 1$.