**abstract:**
We will start from consideration of a random walk on a countable finitely
generated group G. A random walk can either be recurrent or transient. In
case of transiense it is natural to pose a question about the space of
different ways of reaching infinity. Probably the most natural way of
defining such space is as a compactification of G such that a.e. trajectory
of the random walk converges to some point of the compactification. We will
study a mu-boundary, another (weaker) way of defining such space, via a so
called mean-proximal action of G. We will give relevant definitions and
consider the most important examples.
It turnes out that if a group G has a normal hyperbolic subgroup H, then
the corresponding action of a group G on the standard hyperbolic boundary
of H is mean-proximal (i.e. the hyperbolic boundary of H is a mu-boundary
for G). This result is, in essense, a generalization of an earlier result
about mu-boundaries of normal extensions of free groups obtained by A.V.
Malyutin and A.M. Vershik. If time permits, we will discuss some other
results in this direction. The talk is based on a joint work with A.V.
Malyutin.

Tue 4 Dec, 14:00 - 15:00, Sala Conferenze Centro De Giorgi

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