## Hiroshima Mathematical Journal

### Commensurability between once-punctured torus groups and once-punctured Klein bottle groups

Mikio Furokawa

#### Abstract

The once-punctured torus and the once-punctured Klein bottle are topologically commensurable, in the sense that both of them are doubly covered by the twice-punctured torus. In this paper, we give a condition for a faithful type-preserving $\mathrm{PSL}(2,\mathbf{C})$-representation of the fundamental group of the once-punctured Klein bottle to be ‘‘commensurable’’ with that of the once-punctured torus. We also show that such a pair of $\mathrm{PSL}(2,\mathbf{C})$-representations extend to a representation of the fundamental group of a common quotient orbifold. Finally, we give an application to the study of the Ford domains.

#### Article information

Source
Hiroshima Math. J., Volume 46, Number 2 (2016), 217-253.

Dates
Revised: 22 January 2016
First available in Project Euclid: 12 August 2016

https://projecteuclid.org/euclid.hmj/1471024950

Digital Object Identifier
doi:10.32917/hmj/1471024950

Mathematical Reviews number (MathSciNet)
MR3536997

Zentralblatt MATH identifier
1351.57023

#### Citation

Furokawa, Mikio. Commensurability between once-punctured torus groups and once-punctured Klein bottle groups. Hiroshima Math. J. 46 (2016), no. 2, 217--253. doi:10.32917/hmj/1471024950. https://projecteuclid.org/euclid.hmj/1471024950