We establish an exact formula for the word distance on the discrete Heisenberg group ℍ₃ with its standard set of generators. This formula is then used to prove the almost connectedness of the spheres for this distance.
We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.
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