Mirror stabilizers for lattice complex hyperbolic triangle groups

TL;DR

This paper investigates the stabilizers of mirrors in lattice complex hyperbolic triangle groups, providing explicit generators, signatures, and generating pairs for the ambient lattice, advancing understanding of their geometric structure.

Contribution

It introduces explicit generators and signatures for Fuchsian stabilizers and identifies pairs of lines generating the lattice, offering new tools for studying these groups.

Findings

Explicit generators for stabilizers are provided.

Signatures of stabilizers are computed.

Pairs of lines generating the lattice are identified.

Abstract

For each lattice complex hyperbolic triangle group, we study the Fuchsian stabilizer of a reprentative of each group orbit of mirrors of complex reflections. We give explicit generators for the stabilizers, and compute their signature in the sense of Fuchsian groups. For some of the triangle groups, we also find explicit pairs of complex lines such that the union of their stabilizers generate the ambient lattice.