Group generated by half transvections

Abstract

Consider the group SL(2;Z) acting on the circle consisting of rays from the origin in R². The action of parabolic elements or transvections X $\in$ SL(2;Z) (Tr X = 2) have 2 fixed points on the circle. A half transvection is the restriction of the action of a parabolic element to one of the invariant arcs extended by the identity on the other arc. We show that the group G generated by half transvections is isomorphic to the Higman-Thompson group T, which is a finitely presented infinite simple group. A finite presentation of the group T has been known, however, we explain the geometric way to obtain a finite presentation of the group T by the Bass-Serre-Haefliger theory. We also give a finite presentation of the group T by the generators which are half transvections.