Abstract
In 2003 Bieri and Geoghegan generalized the Bieri- Neuman-Strebel invariant � 1 by defining � 1 (�), � an isometric action by a finitely generated group G on a proper CAT(0) space M. In this paper, we show how the natural and well-known connec- tion between Bass-Serre theory and covering space theory provides a framework for the calculation of � 1 (�) whenis a cocompact action by G = B ⋊ A, A a finitely generated group, on a locally finite Bass-Serre tree T for A. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endpoint from, � 1 (�). When A is a finitely generated free group acting on its Cayley graph, we can restate this theorem from a more algebraic perspective, which leads to some general results on � 1 for such actions.
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