M.Sc Thesis
M.Sc Student Shapira Zohar Horospherical Entropy for Actions of Free Groups Department of Mathematics Professor Amos Nevo

Abstract

The entropy theory of decreasing uniform measurable partitions developed originally by Vershik and Stepin is used to define a notion of entropy for probability measure-preserving actions of finitely generated free non-abelian groups. The entropy thus defined for free groups is non-negative, and assumes the expected value for Bernoulli actions. This definition of entropy for actions of free groups is made possible by the construction of the horospherical equivalence relation in such actions.

An alternative geometric process of successive refinement of partitions along horospheres is suggested and studied, and its entropy is computed in some cases. This process is shown to coincide with Vershik's process under some conditions, and this fact is used to establish a Shannon-McMillan mean convergence theorem for the horospherical entropy.

Considering the problem of a pointwise result for general, not necessarily uniform hyperfinite equivalence relations, a modest step is taken towards its solution. However, conditions for the validity of a full Shannon- McMillan-Breiman pointwise convergence theorem are yet to be formulated and proved in this generality.