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Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

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Abstract

We prove a quantitative version of the classical Tits’ alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.

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Notes

  1. Notice that, in this weaker form, the Tits Alternative is no longer a dichotomy for linear groups, since it is well known that there exist solvable groups of \(GL(n,{\mathbb {R}})\) which also contain free semigroups (and actually, any finitely generated solvable group which is not virtually nilpotent contains a free semigroup on two generators [34]). It remains a dichotomy for those classes of groups for which “virtually solvable” implies sub-exponential growth, e.g. hyperbolic groups, groups acting geometrically on CAT(0)-spaces etc.

  2. The authors are not able to understand the proof in [19] without the torsionless assumption, which seems to be used at page 14, below Lemma 4.5.

  3. In [3, 5] the authors only consider compact quotients of convex, \(\delta \)-hyperbolic spaces, under the weaker assumption of bounded entropy; however, the compactness of the class GCB\((P_0,r_0,\delta ; D)\) follows from the same arguments and the contractibility of convex, geodesically bicombed spaces.

  4. Notice that in [2] they use the notation \(\ell (g)\) to denote \(\Vert g \Vert \), while we use \(\ell (g)\) to denote the minimal displacement.

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Acknowledgements

The authors are grateful to G. Besson, G. Courtois and S. Gallot for many helpful discussions.

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Authors and Affiliations

  1. Karlsruhe Institute of Technology, Engelstrasse 2, 76128, Karlsruhe, Germany

    Nicola Cavallucci

  2. Institute of Mathematics, EPFL, Station 8, 1015, Lausanne, Switzerland

    Nicola Cavallucci

  3. Dipartimento di Matematica “Guido Castelnuovo”, SAPIENZA Universita di Roma, Piazzale Aldo Moro 5, 00185, Rome, Italy

    Andrea Sambusetti

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Correspondence to Nicola Cavallucci.

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A. Sambusetti is member of the Differential Geometry section of the GNSAGA-INdAM.

N. Cavallucci is partially supported by the SFB/TRR 191, funded by the DFG.

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Cavallucci, N., Sambusetti, A. Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces. Geom Dedicata 218, 36 (2024). https://doi.org/10.1007/s10711-023-00874-z

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