Complex hyperbolic Kleinian groups of large critical exponents

TL;DR

This paper demonstrates the existence of complex hyperbolic Kleinian groups with critical exponents approaching the maximum, indicating no gap in possible exponent values for these groups.

Contribution

It constructs examples of discrete isometry groups with critical exponents arbitrarily close to the maximum, filling a gap in understanding of their possible exponent values.

Findings

Existence of groups with critical exponents near maximum

No gap in the spectrum of critical exponents for complex hyperbolic Kleinian groups

Critical exponents can be made arbitrarily close to the maximum

Abstract

In this article, we show that there exist discrete isometry groups of the $2$- and $3$-dimensional complex hyperbolic spaces with critical exponents arbitrarily close to but strictly smaller than the maximum possible value. This result shows no gap in the values of critical exponents for complex hyperbolic Kleinian groups.