Affiliation:
1. Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637-1514, USA
2. Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA
Abstract
We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are [Formula: see text]-free for a given integer [Formula: see text]. We show that any such manifold [Formula: see text] contains a point [Formula: see text] with the following property: If [Formula: see text] is the set of maximal cyclic subgroups of [Formula: see text] that contain non-trivial elements represented by loops of [Formula: see text], then for every subset [Formula: see text], we have rank [Formula: see text]. This generalizes to all [Formula: see text] results proved in [J. W. Anderson, R. D. Canary, M. Culler and P. B. Shalen, Free Kleinian groups and volumes of hyperbolic 3-manifolds, J. Differential Geom. 43 (1996) 738–782; M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156] for [Formula: see text]. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136; R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156].
Publisher
World Scientific Pub Co Pte Lt
Subject
Geometry and Topology,Analysis
Cited by
1 articles.
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