The talk is based on two recent preprints :

1. [GGPY], I. Gekhtman, V. Gerasimov, L.P. W. Yang, "Martin boundary covers Floyd boundary" (arXiv:1708.02133),

2. [DGGP], M. Dussaule, I. Gekhtman, V. Gerasimov, L.P. The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups" (arXiv:1711.11307).

We study two different compactifications of finitely generated groups. The first is the Martin compactification which comes from the random walks on the Cayley graph of a group equipped with a symmetric probability measure. The second compactification is the Floyd compactification which is the Cauchy completion of the Cayley graph equipped with a distance obtained by a rescaling of the word metric. The corresponding boundaries are the remainders of the group in these compactifications.

Our first main result from [GGPY] states that the identity map on the group extends to an equivariant and continuous map between Martin and Floyd compactifications. The proof is based on our generalization of the Ancona inequality proved by A. Ancona for hyperbolic groups in 80's.

Using these results we prove in [DGGP] that the Martin boundary of a hyperbolic group *G *relatively to a system of virtually abelian subgroups is a "parabolic blow-up space". It is obtained from the limit set *X *of the relatively hyperbolic action of *G *by replacing every parabolic fixed point *p **∈** X *by the euclidean sphere of dimension *k **− *1 where *k *is the rank of its parabolic stabilizer. All other points of *X *are conical and they remain unchanged.

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