Commentarii Mathematici Helvetici

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Volume 84, Issue 2, 2009, pp. 297–337
DOI: 10.4171/CMH/163

Published online: 2009-06-30

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

Jean-François Lafont[1] and Ivonne J. Ortiz[2]

(1) S.U.N.Y. Binghamton, USA
(2) Miami University, Oxford, United States

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 ℍ3 (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.

Keywords: Classifying spaces, lower algebraic K-theory, Coxeter groups, hyperbolic manifold, Farrel–Jones isomorphism conjecture, relative assembly map, Waldhausen Nil-groups, Farrell Nil-groups, Bass Nil-groups

Lafont Jean-François, Ortiz Ivonne: Lower algebraic K-theory of hyperbolic 3-simplex reflection groups. Comment. Math. Helv. 84 (2009), 297-337. doi: 10.4171/CMH/163