## Kodai Mathematical Journal

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

### Group generated by half transvections

#### Abstract

Consider the group *SL*(2;**Z**) acting on the circle consisting of rays from the origin in **R**^{2}. The action of parabolic elements or transvections *X* *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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 December 2005

**Permanent link to this document**

http://projecteuclid.org/euclid.kmj/1134397761

**Digital Object Identifier**

doi:10.2996/kmj/1134397761

**Mathematical Reviews number (MathSciNet)**

MR2194538

**Zentralblatt MATH identifier**

05031729

#### Citation

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