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On convex hulls and the quasiconvex subgroups of F m ×ℤ n

On convex hulls and the quasiconvex subgroups of F m ×ℤ n Abstract In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of F m ×ℤ n on Tree × ℝ n ${\mathrm {Tree}\times \mathbb {R}^n}$ , every quasiconvex subgroup of F m ×ℤ n is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of F m ×ℤ n are precisely those of the form A × B , where A ≤ F m ${A\le F_m}$ is finitely generated, and B ≤ ℤ n ${B\le \mathbb {Z}^n}$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

On convex hulls and the quasiconvex subgroups of F m ×ℤ n

Groups Complexity Cryptology , Volume 7 (1) – May 1, 2015

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Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc-2015-0006
Publisher site
See Article on Publisher Site

Abstract

Abstract In this paper, we explore a method for forming the convex hull of a subset in a uniquely geodesic metric space due to Brunn and use it to show that with respect to the usual action of F m ×ℤ n on Tree × ℝ n ${\mathrm {Tree}\times \mathbb {R}^n}$ , every quasiconvex subgroup of F m ×ℤ n is convex. Further, we show that the Cartan–Hadamard theorem can be used to show that locally convex subsets of complete and connected CAT(0) spaces are convex. Finally, we show that the quasiconvex subgroups of F m ×ℤ n are precisely those of the form A × B , where A ≤ F m ${A\le F_m}$ is finitely generated, and B ≤ ℤ n ${B\le \mathbb {Z}^n}$ .

Journal

Groups Complexity Cryptologyde Gruyter

Published: May 1, 2015

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