Fundamental polyhedra of projective elementary groups
Daniel E. Martin

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 an imaginary quadratic ring, we compute a fundamental polyhedron of , the projective elementary subgroup of . This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for , show that it has infinite-index and is its own normalizer in , and split into a free product with amalgamation that has as one of its factors.
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Taxonomy
TopicsFinite Group Theory Research · Geometric and Algebraic Topology · Algebraic Geometry and Number Theory
