Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

2015/03/23 by Corey Bregman, Bregman, Corey
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #math.GT

paper · pdf · doi:10.48550/arxiv.1503.06820

31 pages. Added a reference and a remark concerning previous work on the result in Theorem 1.2

openalex publication_date 2015/03/23 · arxiv created 2015/06/03 · arxiv updated 2015/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a matrix A∈ SL(N,\Z), form the semidirect product G=\Z^N\rtimes_A \Z where the \Z factor acts on \Z^N by A. Such a G arises naturally as the fundamental group of an N-dimensional torus bundle which fibers over the circle. In this paper we prove that if A has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup H≤ G possessing rational growth series for some generating set. In contrast, we show that if A has at least one eigenvalue not lying on the unit circle, then G is not almost convex for any generating set.

Citations

• The Geometries of 3-Manifolds

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