Fundamental polyhedra of projective elementary groups

TL;DR

This paper computes a fundamental polyhedron for the projective elementary subgroup of PSL_2 over an imaginary quadratic ring, leading to simplified proofs of key theorems and new structural insights.

Contribution

It provides the first explicit fundamental polyhedron for PE_2(O), enabling new proofs and a detailed structural decomposition of PSL_2(O).

Findings

Derived a presentation for PE_2(O).

Showed PE_2(O) has infinite index and is its own normalizer.

Decomposed PSL_2(O) into a free product with amalgamation.

Abstract

For $O$ an imaginary quadratic ring, we compute a fundamental polyhedron of $\text{PE}_2(O)$, the projective elementary subgroup of $\text{PSL}_2(O)$. This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for $\text{PE}_2(O)$, show that it has infinite-index and is its own normalizer in $\text{PSL}_2(O)$, and split $\text{PSL}_2(O)$ into a free product with amalgamation that has $\text{PE}_2(O)$ as one of its factors.