Title: Permutation representations of the symmetry groups of regular hyperbolic tessellations

Authors: Qaiser Mushtaq and Herman Servatius

Reference: Journal of the London Mathematics Society (2) 48, 77--86, 1993.

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Abstract

It is well known that the modular group,  \Gamma = [ x,y | x^2 = y^3 = 1 ],  has the property that every alternating and symmetric group is a homomorphic image of \Gamma except A_6, A_7, A_8, S_5, S_7, or S_8. Higman has questioned which discrete reflective hyperbolic groups also exhibit this type of behavior. Let G be an infinite, finitely presented group.

We say that G has {\em \propertyh} if there is an positive integer N such that either A_n or S_n is a homomorphic image of G for all n > N. If p and q satisfy (p-2)(q-2) > 4, then {p,q} denotes the tessellation of the hyperbolic plane by regular p--gons, q meeting at each vertex, and [p,q] denotes the symmetry group of that tessellation. It is an infinite Coxeter group generated by the reflections in the sides of the right hyperbolic triangle forming the fundamental region of the tessellation.

It has been shown that [3,q] has \propertyh for all q > 6, see~\cite{Conder2}, and that [4,q] has \propertyh for all q > 6, see~\cite{Mushtaq}. We will show that [p,q] has \propertyh for all for all but perhaps finitely many values of p and q.

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