The real spectrum compactification of character varieties: characterizations and applications

Abstract

The real spectrum compactification of character varieties: characterizations and applications

High energy harmonic maps and degeneration of minimal surfaces

“…is injective. Forthcoming work of Burger, Iozzi, Parreau, and Pozzetti [5] will show the limits of this embedding are given by the projectivization of a pair of measured laminations. The limiting currents thus satisfy i(η, •) = i(λ 1 , •) + i(λ 2 , •), (1.1) where λ 1 and λ 2 are specific representatives of the projectivize classes [λ 1 ] and [λ 2 ], respectively, representing limits on the Thurston boundary.…”

Positive crossratios, barycenters, trees and applications to maximal representations

“…is a vector valued pseudo-distance [BIPP20,§3.4]. Given any Weyl group invariant norm • on R n , δ(Z 1 , Z 2 ) induces a pseudo-distance on X F n , all these pseudo-distances are pairwise equivalent [BIPP21a]. For the purposes of this paper we will only be interested in the pseudo-distance d 1 arising from the ℓ 1 -norm on R n (x 1 , .…”

“…When the real closed field F has an Λ Fvalued valuation v compatible with the ordering, the group PSp(2n, F) acts by isometries on a Λ F -metric space X F n obtained as metric quotient of the non-Archimedean symmetric space (see Section 5.2). The relation between the metric space X F n and the Bruhat-Tits building associated to PSp(2n, F) will be discussed in detail in [BIPP21a], see also [BIPP20,§2.1]. This induces the length function…”

Positive surface group representations in PO(p,q)

“…Respectively, they discovered the Hitchin components for split real Lie groups and the maximal components for Hermitian Lie groups. It was later discovered that these components consist only of representations with further remarkable geometric properties bearing strong similarities with holonomies of hyperbolizations [LZ17, BP17, LM09, VY17, FP20]; furthermore, since they form connected components of character varieties, they can be studied with an array of different tools including Higgs bundles [Hit92,BGPG06] and real algebraic geometry [BIPP21].…”