Graph manifolds with boundary are virtually special
type: Item
instance of: scholarly article
title: Graph manifolds with boundary are virtually special
zbMATH Open document ID: 1331.57023
author: Piotr Przytycki
author: Daniel T. Wise
published in: Journal of Topology
publication date: 2014-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.3513
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 6366502
zbMATH Keywords: graph manifold
zbMATH Keywords: separable subgroup
zbMATH Keywords: special cube complex
zbMATH Keywords: virtually special
MaRDI profile type: Publication
OpenAlex ID: W2103966230
arXiv ID: 1110.3513
DOI: 10.1112/JTOPOL/JTT009
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review text: This paper considers separability properties of surfaces properly embedded in graph manifolds. Recall that a subgroup \(F\) of a group \(G\) is separable if, for each \(g\in G-F\), there is a finite index subgroup \(H\) of \(G\) with \(g\notin HF\). If \(F_1\), \(F_2\) are subgroups of \(G\), the double coset \(F_1F_2\) is separable if, for each \(g\in G-F_1F_2\), there is a finite index subgroup \(H\) of \(G\) with \(g\notin HF_1F_2\). A graph manifold is an oriented compact connected irreducible 3-manifold that has only Seifert-fibred pieces in its \(JSJ\) decomposition. The main results of the paper are the following: a) Let \(M\) be a graph manifold, with or without boundary, and let \(S\) be an oriented incompressible surface properly embedded in \(M\). Then \(\pi_1(S)\) is separable in \(\pi_1(M)\). b) Let \(S, P\subset M\), be oriented incompressible surfaces whose intersection with each block of the \(JSJ\) decomposition of \(M\) is horizontal or vertical. Let \(\tilde S,\tilde P\) be intersecting components of the preimages of \(S,P\) in the universal cover \(\tilde M\) of \(M\). Then \(Stab(\tilde S)Stab(\tilde P)\) is separable in \(\pi_1(M)\).NEWLINENEWLINELet \(M\) be a graph manifold containing a sufficient collection of incompressible oriented surfaces, that is, a collection so that the intersection of each surface with each block of the \(JSJ\) decomposition of \(M\) is vertical or horizontal, and so that for each block \(B\) and each torus in \(\partial B\), there is a surface intersecting them, which is vertical with respect to \(B\). Using the main results it is proved that \(\pi_1(M)\) is virtually the fundamental group of a special cube complex. Then it is shown that a graph manifold with boundary always has a sufficient collection, implying then that if \(M\) is a graph manifold with non-empty boundary, then \(\pi_1(M)\) is virtually special. See [\textit{F. Haglund} and \textit{D. T. Wise}, Geom. Funct. Anal. 17, No. 5, 1551--1620 (2008; Zbl 1…
reviewed by: Mario Eudave-Muñoz
description: scientific article; zbMATH DE number 6366502