Random walks, Kleinian groups, and bifurcation currents

TL;DR

This paper introduces a bifurcation current for holomorphic families of group representations into PSL(2,C), linking random matrix products to bifurcation phenomena and extending concepts from rational dynamics.

Contribution

It defines a new bifurcation current for these families, connecting random matrix theory with bifurcation analysis, and characterizes its support and distribution properties.

Findings

Support of bifurcation current matches bifurcation locus.

Bifurcation current describes distribution of parabolics and relations.

Framework generalizes bifurcation theory to group representations.

Abstract

Let (\rho_\lambda)_{\lambda\in \Lambda} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold \Lambda . We define a notion of bifurcation current in this context, that is, a positive closed current on \Lambda describing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on the Riemann sphere. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure \mu on G. We show that under natural assumptions on \mu, the support of the bifurcation current coincides with the bifurcation locus of the family. We also prove that the bifurcation current describes the asymptotic distribution of several codimension 1 phenomena in parameter space, like accidental parabolics or new relations, or accidental collisions between fixed points.