**Beiträge zur Algebra und Geometrie <BR> Contributions to Algebra and GeometryVol. 40, No. 2, pp. 351-361 (1999)
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Galleries in Poincare Half-spaces and Diophantine Approximation

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Pierrick Meignen

Structures Discretes et Analyse Diophantienne, U.F.R. des Sciences, Universite de Caen, Esplanade de la Paix, 14032 Caen, France, e-mail: meignen@math.unicaen.fr

**Abstract:** \font\msbm=msbm10 \def\bbR{\hbox{\msbm R}} In this paper, we present a multidimensional generalization of a geometrical method of approximation devised by G. Humbert. Introducing Clifford algebras and hyperbolic reflection groups with unbounded fundamental domains of finite volume, we obtain a complete set of results which are traditionally connected with the modular group and the usual continued fraction expansions in $\bbR$.

**Keywords:** Hyperbolic reflection groups, Diophantine approximation

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