We prove that the hitting measure is not equivalent to the Lebesgue measure for a large class of nearest-neighbour random walks on hyperbolic reflection groups and Fuchsian groups with regular Dirichlet polygons.

If A is an algebra with finite right global dimension, then for any automorphism α and α-derivation δ the right global dimension ofWe extend this result to the case of holomorphic Ore extensions and smooth crossed products by Z of ⊗-algebras.

We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.

For an Arens-Michael algebra A we consider a class of A-⊗-bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over A. Given a Fréchet-Arens-Michael algebra A and an topologically invertible Fréchet A-⊗bimodule M , we construct an Arens-Michael algebra L A (M ) which serves as a topological version of the Laurent tensor algebra L A (M ).Also, for a fixed algebra B we provide a condition on an invertible B-bimodule N sufficient for the Arens-Michael envelope of L B (N ) to be isomorphic to L B ( N ). In particular, we prove that the Arens-Michael envelope of an invertible Ore extension A[x, x −1 ; α] is isomorphic to L A ( A α ) provided that the Arens-Michael envelope of A is metrizable.