Growth tightness for groups with contracting elements. (Q2927886)

A finitely generated group \(G\) is growth tight if for every quotient \(\overline G=G/\Gamma\) by an infinite normal subgroup the exponential growth rate of balls in Cayley graphs of \(\overline G\) is strictly smaller than for \(G\). The second main result of the article is that if \(G\) admits a non-trivial Floyd boundary (with respect to some generating set and Floyd function) then it is growth tight.NEWLINENEWLINE If a group acts properly on a proper geodesic metric space, it inherits a pseudometric from that action and growth tightness can be expressed with respect to this metric (taking for the quotients the metric inherited from the action on the quotient space). The first main result of the article is that if a group that is not virtually cyclic acts geometrically on a proper geodesic metric space and if the action admits a contracting element then the group is growth tight with respect to that action. The second main result is deduced from the first.NEWLINENEWLINE An element \(h\in G\) is contracting if the projection of a quasi-geodesic to the group \(\langle h\rangle\) has uniformly small diameter (depending on the distance between the quasi-geodesic and \(\langle h \rangle\)).NEWLINENEWLINE Each of these properties represents a kind of negative curvature phenomenon. For example, it follows directly from the first main theorem that hyperbolic groups are growth tight (which was known before [\textit{S. Sabourau}, J. Mod. Dyn. 7, No. 2, 269-290 (2013; Zbl 1286.20057)]) and that CAT(\(0\)) groups with a rank-\(1\) element are growth tight (using that rank-\(1\) elements are contracting by \textit{M. Bestvina} and \textit{M. Feighn} [J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029); addendum and correction ibid. 43, No. 4, 783-788 (1996; Zbl 0862.57027)]). Since relatively hyperbolic groups have non-trivial Floyd boundary by \textit{V. Gerasimov} [Geom. Funct. Anal. 22, No. 5, 1361-1399 (2012; Zbl 1276.20050)], it follows from the second main theorem that they are growth tight as well (generalizing [\textit{G. N. Arzhantseva} and \textit{I. G. Lysenok}, Math. Z. 241, No. 3, 597-611 (2002; Zbl 1045.20035)] and [\textit{A. Sambusetti}, in: Recent advances in geometry and topology. Proceedings of the 6th international workshop on differential geometry and its applications and the 3rd German-Romanian seminar on geometry, Cluj-Napoca, Romania, September 1-6, 2003. Cluj-Napoca: Cluj University Press. 341-352 (2004; Zbl 1086.53055)]). The author also notes that groups with non-trivial Floyd boundary contain hyperbolically embedded subgroups.NEWLINENEWLINE The proof of the first main theorem proceeds as follows: it is shown that if \(G\) contains a contracting element and \(\Gamma\) is an infinite normal subgroup then \(\Gamma\) contains a contracting element \(h\). Further there is a subset \(A\) of a set of coset representatives such that the free product of sets \(A*h\) injects into \(G\). Since the growth rate of \(A*h\) is strictly larger than that of \(A\) the result follows.NEWLINENEWLINE The claims used above are proven using geometric methods. The main geometric statement shows that certain piecewise quasi-geodesics (so-called admissible paths) are quasi-geodesics. For example, if \(\gamma\) is the composition of a geodesic \(p\) that is contained in a set \(X\) (from a contracting system) and a quasi-geodesic \(q\) that has bounded projection to \(X\) then \(\gamma\) is a quasi-geodesic (with constants depending on the constants for the contracting system, the quasi-geodesic and the bounded projection).