Analogs of complementary series for CAT(-1) groups

TL;DR

This paper generalizes the construction of complementary series representations to a broader class of groups acting on CAT(-1) spaces, extending known methods from classical negatively curved groups to new, non-linear examples.

Contribution

It introduces a dynamical approach to extend complementary series representations to convex-cocompact CAT(-1) groups, including non-linear groups from hyperbolic buildings.

Findings

Constructed new complementary series representations for CAT(-1) groups.

Generalized known constructions from classical groups to hyperbolic building lattices.

Provided a dynamical framework applicable to a wider class of negatively curved groups.

Abstract

In this paper we extend the construction of complementary series representations to convex-cocompact isometry groups of CAT(-1) spaces with conditionally negative metrics. Our approach is purely dynamical and generalizes the constructions known for negatively curved algebraic groups as $SO(n,1)$, $SU(n,1)$ or $SL_2(\mathbb{Q}_p)$ and their lattices to new examples as non-linear groups coming from lattices of certain hyperbolic buildings.