, Then, there exists a real number ? G such that for any loxodromic element g, l G (g) ? ? G N

A. Ancona, Positive harmonic functions and hyperbolicity, Potential theory-surveys and problems, pp.1-23, 1988.

S. Blachère and S. Brofferio, Internal diffusion limited aggregation on discrete groups having exponential growth, Probabability Theory and Related Fields, vol.137, pp.323-343, 2007.

S. Blachère, P. Haïssinsky, and P. Mathieu, Asymptotic entropy and Green speed for random walks on countable groups, The Annals of Probability, vol.36, pp.1134-1152, 2008.

S. Blachère, P. Haïssinsky, and P. Mathieu, Harmonic measures versus quasiconformal measures for hyperbolic groups, Annales Scientifiques de l'École Normale Supérieure, vol.44, pp.683-721, 2011.

B. Bowditch, Geometrical finiteness with variable negative curvature, Duke Mathematical Journal, vol.77, pp.229-274, 1995.

B. Bowditch, Convergence groups and configuration spaces, Group theory down under, pp.23-54, 1999.

B. Bowditch, Relatively hyperbolic group, International Journal of Algebra and Computation, vol.22, p.66, 2012.

E. Candellero, L. Gilch, and S. Müller, Branching random walks on free products of groups, Proceedings of the London Mathematical Society, vol.104, pp.1085-1120, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01255386

M. Coornaert, Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov (French), Pacific Journal of Mathematics, vol.159, pp.241-270, 1993.

F. Dal and &. Bo, Remarques sur le spectre des longueurs d'une surface et comptages (French), vol.30, pp.199-221, 1999.

T. Delzant, Sous-groupes distingués et quotients des groupes hyperboliques (French), Duke Mathematical Journal, vol.83, pp.661-682, 1996.

Y. Derriennic, Entropie, théorèmes limites et marches aléatoires (French), Probability Measures on Groups VIII, pp.241-284, 1986.

C. Dru?u and M. Sapir, Tree graded spaces and asymptotic cones of groups, Topology, vol.44, pp.959-1058, 2005.

M. Dussaule, The Martin boundary of a free product of abelian groups, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01588480

M. Dussaule, I. Gekhtman, V. Gerasimov, and L. Potyagailo, The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01652248

B. Farb, Relatively hyperbolic groups. Geometric and Functional Analysis, vol.8, pp.810-840, 1998.

H. Federer, Geometric Measure Theory. Classics in Mathematics, 1996.

A. Furman, Coarse-geometric perspective on negatively curved manifolds and groups, Rigidity in Dynamics and Geometry, pp.149-166

. Springer, , 2002.

I. Gekhtman, V. Gerasimov, L. Potyagailo, and W. Yang, Martin boundary covers Floyd boundary, 2017.

I. Gekhtman, S. Taylor, and G. Tiozzo, Counting problems in graph products and relatively hyperbolic groups, 2017.

I. Gekhtman and G. Tiozzo, Stationary measures vs Gibbs measures for geometrically finite actions on CAT(-1) spaces, 2019.

V. Gerasimov, Expansive convergence groups are relatively hyperbolic. Geometric and Functional Analysis, vol.19, pp.137-169, 2009.

V. Gerasimov, Floyd maps for relatively hyperbolic groups. Geometric and Functional Analysis, vol.22, pp.1361-1399, 2012.

V. Gerasimov and L. Potyagailo, Quasi-isometries and Floyd boundaries of relatively hyperbolic groups, Journal of the European Mathematical Society, vol.15, pp.2115-2137, 2013.

V. Gerasimov and L. Potyagailo, Non-finitely generated relatively hyperbolic groups and Floyd quasiconvexity. Groups, Geometry, and Dynamics, vol.9, pp.369-434, 2015.

V. Gerasimov and L. Potyagailo, Quasiconvexity in the relatively hyperbolic groups, Journal for Pure and Applied Mathematics, vol.710, pp.95-135, 2016.

S. Gouëzel, Frédéric Mathéus, and François Maucourant. Entropy and drift in word hyperbolic groups. Inventiones Mathematicae, vol.211, pp.1201-1255, 2018.

M. Gromov, Hyperbolic groups, Essays in group theory, vol.8, pp.75-265, 1987.

Y. Guivarc and &. , Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire (French), Conference on Random Walks, vol.74, pp.47-98, 1980.

P. Haïssinsky, Regards croisés sur les marches aléatoires et la géométrie des groupes, 2011.

G. Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups. Algebraic and Geometric Topology, vol.10, pp.1807-1856, 2010.

V. Kaimanovich, Boundaries of invariant Markov operators: the identification problem, Ergodic theory of Z d actions, vol.228, pp.127-176, 1996.

V. Kaimanovich and A. Vershik, Random walks on discrete groups: boundary and entropy, Annals of Probability, vol.11, pp.457-490, 1983.

A. Vadim and . Kaimanovich, The Poisson formula for groups with hyperbolic properties, Annals of Mathematics, vol.152, pp.659-692, 2000.

A. Karlsson, Boundaries and random walks on finitely generated infinite groups, Arkiv för Matematik, vol.41, pp.295-306, 2003.

A. Karlsson, Free subgroups of groups with nontrivial Floyd boundary, Communications in Algebra, vol.31, pp.5361-5376, 2003.

F. Ledrappier, Some asymptotic properties of random walks in free groups, Topics in Probability and Lie Groups: Boundary Theory, pp.117-152, 2001.

J. Maher and G. Tiozzo, Random walks on weakly hyperbolic groups, Journal für die reine und angewandte Mathematik, vol.742, pp.187-239, 2018.

A. Mal and &. Cev, On a class of homogeneous spaces, American Mathematical Society Translations, vol.39, 1951.

P. Mathieu and A. Sisto, Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01271743

K. Matsuzaki, Y. Yabuki, and J. Jaerisch, Normalizer, divergence type and Patterson measure for discrete groups of the Gromov hyperbolic space, 2015.

B. Nica and J. ?pakula, Strong hyperbolicity, 2014.

D. Osin, Acylindrically hyperbolic groups, Transactions of the American Mathematical Society, vol.368, pp.851-888, 2016.

J. Otal, Sur la géométrie symplectique de l'espace des géodésiques d'une variété à courbure négative (French). Revista matematica Ibero americana, vol.8, 1992.

P. Pansu, Croissance des boules et des géodésiques fermées dans les nivariétés (French). Ergodic Theory and Dynamical Systems, vol.3, pp.415-445, 1983.

L. Potyagailo and W. Yang, Hausdorff dimension of boundaries of relatively hyperbolic groups, 2016.

. Madabusi-santanam-raghunathan, Discrete subgroups of Lie groups, 1972.

S. Sawyer, Martin boundaries and random walks. Contemporary mathematics, vol.206, pp.17-44, 1997.

A. Sisto, On metric relative hyperbolicity, 2012.

R. Tanaka, Dimension of harmonic measures in hyperbolic spaces. Ergodic Theory and Dynamical Systems, vol.39, pp.474-499, 2019.

P. Tukia, Conical limit points and uniform convergence groups, Journal for Pure and Applied Mathematics, vol.501, pp.71-98, 1998.

A. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples. Uspekhi Matematicheskikh Nauk, vol.55, pp.59-128, 2000.

C. Walsh, The action of a nilpotent group on its horofunction boundary has finite orbits. Groups Geometry, and Dynamics, vol.5, pp.189-206, 2011.

P. Walters, An Introduction to Ergodic Theory, 1982.

W. Woess, Random Walks on Infinite Graphs and Groups, 2000.

A. Yaman, A topological characterisation of relatively hyperbolic groups, Journal for Pure and Applied Mathematics, vol.566, pp.41-89, 2004.

W. Yang, Patterson-Sullivan measures and growth of relatively hyperbolic groups, 2013.