Vol. 255, No. 1, 2012

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Connectivity properties for actions on locally finite trees

Keith Jones

Vol. 255 (2012), No. 1, 143–154
Abstract

Given an action G↷ρT by a finitely generated group on a locally finite tree, we view points of the visual boundary ∂T as directions in T and use ρ to lift this sense of direction to G. For each point E ∂T, this allows us to ask whether G is (n1)-connected “in the direction of E.” Then the invariant Σn(ρ) ∂T records the set of directions in which G is (n 1)-connected. We introduce a family of actions for which Σ1(ρ) can be calculated through analysis of certain quotient maps between trees. We show that for actions of this sort, under reasonable hypotheses, Σ1(ρ) consists of no more than a single point. By strengthening the hypotheses, we characterize precisely when a given end point lies in Σn(ρ) for any n.

Keywords
BNSR invariant, controlled connectivity, Bieri–Geoghegan invariant, trees, finiteness properties, boundary at infinity
Mathematical Subject Classification 2010
Primary: 20E08, 20F65
Milestones
Received: 29 April 2011
Revised: 28 September 2011
Accepted: 10 October 2011
Published: 14 March 2012
Authors
Keith Jones
Department of Mathematics
Trinity College
300 Summit Street
Hartford, CT 06106
United States