Linear Bounds of Degrees of Alternating Quotients of the (3, q, r) Triangle Groups

Abstract

In Everitt [Everitt, B. J. (2000). Alternating quotients of Fuchsian groups. J. Algebra 223: 457–476], it was shown, in particular, that each Fuchsian triangle group Δ(p, q, r) has among its homomorphic images all but finitely many of the alternating groups. Treating p, q as fixed, the methods of Everitt (2000) give a quadratic function N(r) of r such that A _n is an image of Δ(p, q, r) for every integer n ≥ N(r). We conjecture that there is a linear function of r with this property. In this paper, we will show that the conjecture holds for the Fuchsian triangle groups Δ(3, q, r).

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Acknowledgment

The author is very grateful to Professor John S. Wilson for his generous help and encouragement.

Notes

^aVery recently, Martin Liebeck and Aner Shalev have proved the Higman conjecture using some completely new (non-constructive) ideas (see Liebeck and Shalev, to appear).

^#Communicated by P. Higgins.