Geodesic
Examples of non-abelian discrete groups with non-trivial exit boundary
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part V, Tome 123 (1983), pp. 167-184

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Hew types of counterexamples are given disproving some old conjectures on relations between exit (or, Poisson) boundary of a random walk on a group and its amenability and growth. Random walk with non-trivial exit boundaries on the affine group of the diadic-rational line and on the infinite symmetric group are constructed. 
@article{ZNSL_1983_123_a12,
     author = {V. A. Kaimanovich},
     title = {Examples of non-abelian discrete groups with non-trivial exit boundary},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {167--184},
     publisher = {mathdoc},
     volume = {123},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1983_123_a12/}
}
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V. A. Kaimanovich. Examples of non-abelian discrete groups with non-trivial exit boundary. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part V, Tome 123 (1983), pp. 167-184. http://geodesic.mathdoc.fr/item/ZNSL_1983_123_a12/