Homological Stability for Spaces of Commuting Elements in Lie Groups

On Spaces of Commuting Elements in Lie Groups

The purpose of this paper is to introduce a new method of “stabilizing” spaces of homomorphisms Hom(π,G) where π is a certain choice of finitely generated group and G is a compact Lie group. The main results apply to the space of all ordered n-tuples of pairwise commuting elements in a compact Lie group G, denoted Hom(Zn, G), by assembling these spaces into a single space for all n ≥ 0. The res...

Almost commuting elements in compact Lie groups

Mirror symmetry, Langlands duality, and commuting elements of Lie groups

By normalizing a component of the space of commuting pairs of elements in a reductive Lie group G, and the corresponding space for the Langlands dual group, we construct pairs of hyperkähler orbifolds which satisfy the conditions to be mirror partners in the sense of Strominger-Yau-Zaslow. The same holds true for commuting quadruples in a compact Lie group. The Hodge numbers of the mirror partn...

Homological Stability for Automorphism Groups

We prove a general homological stability theorem for families of automorphism groups in certain categories. We show that this theorem can be applied to all the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups, and obtain new stability theorems with twisted coefficients for the braid groups, automorphisms of free groups...

Homological Stability for automorphism groups of Raags

The stabilization of Hi(Aut(G);Z) for i large has been shown to hold for most groups by the main theorem of [3] and [9, Corollary 1.3].1 The stabilization of Hi(Aut(G);Z) in contrast has so far only been known in two extreme cases: when G is abelian and when G has trivial center and does not factorize as a direct product. Indeed, in the first case Aut(Gn) is isomorphic to GLn(End(G)), which is ...