Fundamental polyhedra of projective elementary groups
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Daniel E. Martin
Abstract
For đť“ž an imaginary quadratic ring, we compute a fundamental polyhedron in hyperbolic 3-space for the action of PE2(đť“ž), the projective elementary subgroup of PSL2(đť“ž). This allows for new, simplified proofs of theorems of Cohn, Nica, Fine, and Frohman. Namely, we obtain a presentation for PE2(đť“ž), show that it has infinite index and is its own normalizer in PSL2(đť“ž), and split PSL2(đť“ž) into a free product with amalgamation that has PE2(đť“ž) as one of its factors.
Funding statement: The author’s research is supported by NSF grant 2336000.
Acknowledgements
The author is grateful to John Cremona for providing the data used to make Figure 1 and to the anonymous reviewers whose comments have improved the paper’s clarity.
Communicated by: T. Grundhöfer
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Articles in the same Issue
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
