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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.
##### MSC:
 51M10 Hyperbolic and elliptic geometries (general) and generalizations 20H10 Fuchsian groups and their generalizations (group-theoretic aspects) 57S30 Discontinuous groups of transformations