On volumes of hyperbolic Coxeter polytopes and quadratic forms

2012/03/29 by John G. Ratcliffe, Ratcliffe, John G., Steven T. Tschantz +1
Mathematics · Physics and Astronomy · #11E45 #20F55 #51F15 #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1203.6648

openalex publication_date 2012/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we compute the covolume of the group of units of the quadratic form f_dⁿ(x) = x₁² + x₂² + . . . + x_n² - d x_n+1² with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f₃ⁿ. We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.

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