Hiroshima Mathematical Journal

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

Mikio Furokawa

Full-text: Open access

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
Received: 19 November 2015
Revised: 22 January 2016
First available in Project Euclid: 12 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1471024950

Digital Object Identifier
doi:10.32917/hmj/1471024950

Mathematical Reviews number (MathSciNet)
MR3536997

Zentralblatt MATH identifier
1351.57023

Subjects
Primary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 57M50: Geometric structures on low-dimensional manifolds

Keywords
Jorgensen theory once-punctured torus once-punctured Klein bottle Ford domain

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


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