Kodai Mathematical Journal

Group generated by half transvections

Takashi Tsuboi

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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.

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Kodai Math. J. Volume 28, Number 3 (2005), 463-482.

First available in Project Euclid: 12 December 2005

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Tsuboi, Takashi. Group generated by half transvections. Kodai Math. J. 28 (2005), no. 3, 463--482. doi:10.2996/kmj/1134397761. https://projecteuclid.org/euclid.kmj/1134397761

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