# Rank of a group generated by side-pairing isometries of a polyhedron

Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?

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I assume that you want $P$ to be a fundamental domain for $G$. Then the answer is positive, see:
Even for cyclic groups, the number of faces of convex (Dirichlet) fundamental polyhedron could be arbitrarily high, see T. Drumm, J. Poritz, Ford and Dirichlet fundamental domains for cyclic subgroups of $PSL(2; C)$, Conformal Geometry and Dynamics, 3 (1999) 116-150. Short of taking a group generated by reflections, I do not think there are nice reasonably general conditions to ensure a relation between rank and the number of faces. – Misha Mar 5 '12 at 3:38