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An infinite family of regular tessellations of \(H^3\). (English) Zbl 0993.51007
In the very precisely written paper the authors introduce the infinite family of groups \[ \Gamma_{m,n}= \bigl\langle s,t,r\mid s^m=(t^{-1}r)^n= (rts)^2= (rsr)^2=[s,t]=1 \bigr\rangle \] and prove the two following theorems:
1. For any \(m,n>2\), there is a faithful representation of \(\Gamma_{m,n}\) in PSL\((2, C)\).
2. The group \(\Gamma_{m,n}\) acts as a discontinuous cocompact group of hyperbolic motions on the hyperbolic space \(H^3\) for any \(m,n>2\).
Furthermore they describe a fundamental domain of \(\Gamma_{m,n}\) and the corresponding metric properties.
51M10 Hyperbolic and elliptic geometries (general) and generalizations
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
57S30 Discontinuous groups of transformations