Acylindrically hyperbolic groups (Q2796509)

Let \(G\) be a group and \((X,d)\) a metric space. The action of \(G\) on \(X\) is called acylindrical if for every \(\varepsilon > 0\) there exists \(R, N >0\) such that for every two points \(x,y \in X\) with \(d(x,y) \geq R\), there are at most \(N\) elements \(g \in G\) satisfying \(d(x,gx) \leq \varepsilon\) and \(d(y,gy) \leq \varepsilon\).NEWLINENEWLINEAn element \(g \in G\) of a group \(G\) acting on a hyperbolic space \(S\) is called loxodromic if the map \(\mathbb{Z} \to S\) defined by \(n \mapsto g^ns\) is a quasi-isometry for some \(s \in S\).NEWLINENEWLINEThe main result of the article is the following theorem:NEWLINENEWLINEFor any group \(G\), the following conditions are equivalent: {\parindent=12mm \begin{itemize}\item[(AH] There exists a generating set \(X\) of \(G\) such that the corresponding Cayley graph \(\Gamma(G, X)\) is hyperbolic, \(|\partial \Gamma(G,X)| > 2\), and the natural action of \(G\) on \(\Gamma(G,X)\) is acylindrical. \item[(AH] \(G\) admits a non-elementary acylindrical action on a hyperbolic space. \item[(AH] \(G\) is not virtually cyclic and admits an action on a hyperbolic space such that at least one element of \(G\) is loxodromic and satisfies the Bestvina-Fujiwara weak proper discontinuity condition as introduced in [\textit{M. Bestvina} and \textit{K. Fujiwara}, Geom. Topol. 6, 69--89 (2002; Zbl 1021.57001)]. \item[(AH] \(G\) contains a proper infinite hyperbolically embedded subgroup. NEWLINENEWLINE\end{itemize}} The implications (AH\(_{1}\)) \(\implies\)(AH\(_{2}\)) \(\implies\) (AH\(_{3}\)) follow from the definitions and (AH\(_{3}\)) \(\implies\)(AH\(_{4}\)) was proved in [\textit{F. Dahmani} et al., Mem. Am. Math. Soc. 1156, 149 p. (2016; Zbl 1396.20041)]. The article in review proves the remaining implication. The article then also shows another similar theorem which allows the author to define a natural notion of a generalized loxodromic element of a group.