Triangle group representations and regular maps
by
Jozef Širáň
Slovak Technical University

Orientable maps whose group of all orientation-preserving automorphisms acts transitively on darts (edges with direction) are known as regular maps. On a group-theoretical level, regular maps correspond to normal subgroups of the triangle groups T(m, n, 2)= < u, v| u^m=vⁿ = (uv)²=1 > . Rather than working with T(m, n, 2) as abstract groups, it is sometimes easier to identify certain normal subgroups (and hence regular maps) by means of suitable faithful representations of T(m, n, 2).

If 1/m+1/n < 1/2, the triangle group T(m, n, 2) acts as a group of hyperbolic isometries on the universal tessellation of a hyperbolic plane by n-gons, n of them at each vertex. Referring to standard models of a hyperbolic plane, we will revisit two basic methods of constructing faithful representations of triangle groups in special linear groups over certain polynomial rings. Such representations naturally lead to the study of regular maps that correspond to congruence subgroups of linear groups; we will present applications and discuss limitations of this "congruence subgroup technique".

Date received: November 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafp-33.