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**Random walk in random groups.**
*(English)*
Zbl 1122.20021

From the introduction: This paper compiles basic components of the construction of random groups and of the proof of their properties announced in [M. Gromov, Spaces and questions, GAFA 2000, Geom. Funct. Anal., Special Volume, Basel: Birkhäuser, 118-161 (2000; Zbl 1006.53035)]. Justification of each step, as well as the interrelation between them, is straightforward by available techniques specific to each step. On the other hand, there are several ingredients that cannot be truly appreciated without extending the present framework. We shall indicate along the way possible developments postponing full exposition to forthcoming articles expanding the following points touched upon in the present paper. I. Notions of randomness inside and outside infinite groups. II. Small cancellation theories for rotation families of groups. III. Diffusion, codiffusion, relaxation constants and Kazhdan’s T. IV. Entropies of random walks, Hausdorff-Gibbs limit of mm spaces, and mean hyperbolicity. V. Non-geodesic metric spaces, Gibbs’ hulls and fractal hyperbolicity. VI. Entropies of displacements. VII. Families of expanders.

### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20F65 | Geometric group theory |

20P05 | Probabilistic methods in group theory |

60G50 | Sums of independent random variables; random walks |

20F06 | Cancellation theory of groups; application of van Kampen diagrams |

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |