Geometric limits of cyclic subgroups of SO_0(1, k+1) and SU(1, k+1)

TL;DR

This paper investigates the geometric limits of cyclic subgroups in certain Lie groups, providing new examples, conditions for limits, and applications to free group representations.

Contribution

It constructs new examples of geometric limits, establishes criteria for subgroup limits, and extends the analysis to free group representations.

Findings

Sequences of cyclic subgroups can have geometric limits larger than algebraic limits.

Necessary and sufficient conditions for a subgroup to be a geometric limit.

Applications to sequences of free group representations.

Abstract

We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of subgroups of such groups G that converge algebraically and whose geometric limit strictly contains the algebraic limit, thus generalizing the example first described by Jorgensen for subgroups of SO_0(1,3). We also give necessary and sufficient conditions for a subgroup of SO_0(1, k+1) to arise as geometric limit of a sequence of cyclic subgroups. We then discuss generalizations of such examples to sequence of representations of free groups, and applications of our constructions in that setting.