Boundary behavior for groups of subexponential growth

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Milnor's Problem on the Growth of Groups and its Consequences (2011, methodology): “To obtain a lower bound for growth for some groups from the family Gω and to obtain new results about the Poisson-Furstenberg boundary of random walks on groups of intermediate growth, in [73] Erschler introduced the so called “strong condition” (∗) for some type actions on the interval (0, 1].”
Growth of permutational extensions (2010, background): “It is shown in [11] that a there exist groups of intermediate growth which admit a non-degenerate measure with non-trivial Poisson-Furstenberg boundary.”
Asymptotic aspects of Schreier graphs and Hanoi Towers groups (2006, background): “For example, γ (n) ∼ n log 2 ( 3 ) in case of H ( 3 ) , and the growth is linear for the ﬁrst group from [10] and the Erschler group [8].”
Hausdorff dimension in a family of self-similar groups (2006, background): “This group is studied in more detail in [7].”
Groups of given intermediate word growth (2011, methodology): “On the one hand, there are combinatorial constructions of sufficiently many elements in the ball of radius R, see [2,11,24]; probabilistic methods have also been used, see [13, 15].”
Poisson-Furstenberg Boundaries, Large-scale Geometry and Growth of Groups (2011, methodology): “A negative answer is given in [32], where it is shown that some among Grigorchuk groups of intermediate growth admit a measure with non-trivial boundary.”