Lower algebraic $K$-theory of hyperbolic 3-simplex reflection groups

Abstract

A hyperbolic 3-simplex reflection group is a Coxeter group arising as a lattice in $O^{+}(3,1)$, with fundamental domain a geodesic simplex in $H^3$ (possibly with some ideal vertices). The classification of these groups is known, and there are exactly 9 cocompact examples, and 23 non-cocompact examples. We provide a complete computation of the lower algebraic $K$-theory of the integral group ring of all the hyperbolic 3-simplex reflection groups.

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