Group generated by half transvections. (English) Zbl 1096.20032
The group \(\text{SL}(2,\mathbb{Z})\) acts on the circle consisting of rays from the origin in \(\mathbb{R}^2\). Every nontrivial parabolic element (transvection) has 2 fixed points. The restriction of the action to one of the invariant arcs extended by the identity on the other arc is called a half transvection. It is shown that the group \(G\) generated by half transvections is isomorphic to the Higman-Thompson group \(T\) which was known to be a finitely presented infinite simple group. Using the isomorphism, new finite presentations of \(T\) are obtained.
Reviewer: L. N. Vaserstein (University Park)
MSC:
20F05 | Generators, relations, and presentations of groups |
57S25 | Groups acting on specific manifolds |
20E32 | Simple groups |
57S30 | Discontinuous groups of transformations |