On distortion of normal subgroups
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On distortion of normal subgroups Hung Cong Tran1 Received: 26 September 2017 / Accepted: 28 July 2020 © Springer Nature B.V. 2020
Abstract We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the distortion of normal subgroups by decomposing the whole group into smaller subgroups. We apply our work to compute the distortion of normal subgroups of graph of groups and normal subgroups of right-angled Artin groups that induce infinite cyclic quotient groups. We construct normal subgroups of CAT(0) groups introduced by Macura and introduce a collection of normal subgroups of right-angled Artin groups. These groups provide a rich source to study the connection between subgroup distortion and group divergence on CAT(0) groups. Keywords Normal subgroups · Subgroup distortion · Group divergence Mathematics Subject Classification (2000) 20F67 · 20F65
1 Introduction Subgroup distortion is a famous tool to study the geometric connection between a finitely generated group and its finitely generated subgroups. A metric on a subgroup can be inherited from the metric of the whole group. Also a finitely generated subgroup can be a metric space itself. Therefore, the subgroup distortion notion was introduced to measure the difference between the two metric structures of a finitely generated subgroup. In this article, we only focus on distortion of finitely generated normal subgroups. The divergence is a quasi-isometry invariant which arose in the study of non-positively curved manifolds and metric spaces. Roughly speaking, the divergence of a connected metric space measures the complement distance of a pair of points in a sphere as a function of the radius. For example, the divergence of the Euclidean plane is linear and the divergence of the hyperbolic plane is exponential. The concept of divergence on finitely generated groups is defined via their Cayley graphs. Connection between group divergence and distortion of normal subgroups was first shown by Gersten [6]. He proved the distortion of a finitely generated normal subgroup H in a finitely
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Hung Cong Tran [email protected] Department of Mathematics, The University of Georgia, 1023 D. W. Brooks Drive, Athens, GA 30605, USA
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Geometriae Dedicata
generated G is bounded below by the divergence of G when the quotient group G/H is an infinite cyclic group. We generalize the result of Gersten by the following theorem: Theorem 1.1 Let G be a finitely generated group and H a finitely generated infinite index infinite normal subgroup of G. Then the divergence of G is dominated by the subgroup distortion of H in G. By the above theorem, we can use the divergence of the whole group as a lower bound for distortion of any finitely generated normal subgroup. Also, we can use subgroup distortion as a tool to compute the upper bound for group divergence. In this paper, we revisit Macura’s examples on CAT(0) groups with polynomial divergence (see [8]) and show an al
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