تقرير
Hyperbolic structures on groups
العنوان: | Hyperbolic structures on groups |
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المؤلفون: | Abbott, Carolyn, Balasubramanya, Sahana, Osin, Denis |
المصدر: | Algebr. Geom. Topol. 19 (2019) 1747-1835 |
سنة النشر: | 2017 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Group Theory, 20F65, 20F67, 20E08 |
الوصف: | For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $\mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic; two generating sets of $G$ are equivalent if the corresponding word metrics on $G$ are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded $G$-actions on hyperbolic spaces. We are especially interested in the subset $\mathcal{AH}(G)\subseteq \mathcal{H}(G)$ of acylindrically hyperbolic structures on $G$, i.e., hyperbolic structures corresponding to acylindrical actions. Elements of $\mathcal{H}(G)$ can be ordered in a natural way according to the amount of information they provide about the group $G$. The main goal of this paper is to initiate the study of the posets $\mathcal{H}(G)$ and $\mathcal{AH}(G)$ for various groups $G$. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of $G$, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths. |
نوع الوثيقة: | Working Paper |
DOI: | 10.2140/agt.2019.19.1747 |
URL الوصول: | http://arxiv.org/abs/1710.05197 |
رقم الأكسشن: | edsarx.1710.05197 |
قاعدة البيانات: | arXiv |
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