Commensurability between once-punctured torus groups and once-punctured Klein bottle groups
type: Item
instance of: scholarly article
title: Commensurability between once-punctured torus groups and once-punctured Klein bottle groups
zbMATH Open document ID: 1351.57023
author: Mikio Furokawa
published in: Hiroshima Mathematical Journal
publication date: 2016-10-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hmj/1471024950
reviewed by: Luisa Paoluzzi
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 51M10
zbMATH DE Number: 6643506
zbMATH Keywords: once-punctured torus
zbMATH Keywords: once-punctured Klein bottle
zbMATH Keywords: Ford domain
zbMATH Keywords: commensurability
zbMATH Keywords: faithful and type-preserving representations
MaRDI profile type: Publication
OpenAlex ID: W2493872385
Wikidata QID: Q128870649
DOI: 10.32917/HMJ/1471024950
review text: The fundamental groups of the once-punctured torus and Klein bottle are both free of rank \(2\), and so abstractly commensurable. In this case, they are also \textit{topologically commensurable}, in the sense that both surfaces are double-covered by the twice-punctured torus. On the other hand, if the two surfaces are endowed with a hyperbolic structure, one cannot expect them to have a finite common geometric cover in general. This can be restated by saying that if one considers a discrete and faithful representation in the isometry group of the hyperbolic plane of the once-punctured torus, so that the element representing the puncture is parabolic, and similarly for the once-punctured Klein bottle, the images of the two representations need not be commensurable (up to conjugacy) in the isometry group of the hyperbolic plane.NEWLINENEWLINEHere, the related question of understanding when faithful and type-preserving (i.e., where elements corresponding to cusps are mapped to parabolics) representations of the two groups can be commensurable in \(\mathrm{PSL}(2,\mathbb C)\) is considered. Note that the isometry group of the hyperbolic plane is naturally a subgroup of \(\mathrm{PSL}(2,\mathbb C)\).NEWLINENEWLINEMore precisely, the author provides a condition that ensures that, given a faithful and type preserving representation of the fundamental group of the once-punctured Klein bottle, there is a commensurable faithful and type-preserving representation of the fundamental group of the once-punctured torus.NEWLINENEWLINEA main step in the proof of the result is to show the following two facts: (1) Every type-preserving representation of the fundamental group of the once-punctured torus (resp. once-punctured Klein bottle) extends to a unique representation of an extension of the group corresponding to a double covering of an orbifold by the surface; moreover the representation of the group is faithful if and only if so is the representation of its extension. (2) The re…
description: scientific article; zbMATH DE number 6643506