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# Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2018: 0.38

Online
ISSN
1869-6104
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Volume 7, Issue 1

# On convex hulls and the quasiconvex subgroups of Fm×ℤn

Jordan Sahattchieve
Published Online: 2014-04-15 | DOI: https://doi.org/10.1515/gcc-2015-0006

## 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 Fm×ℤn on $\mathrm{Tree}×{ℝ}^{n}$, every quasiconvex subgroup of Fm×ℤn 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 Fm×ℤn are precisely those of the form A×B, where $A\le {F}_{m}$ is finitely generated, and $B\le {ℤ}^{n}$.

Keywords: CAT(0); quasiconvex; convex hull

MSC: 20F65; 20F67

Published Online: 2014-04-15

Published in Print: 2015-05-01

Citation Information: Groups Complexity Cryptology, Volume 7, Issue 1, Pages 69–80, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144,

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