Amenability and the Liouville property

Amenability and the Liouville Property

We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid actio...

Relative property A and relative amenability for countable groups

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if G has property A relative to a family of subgroups H and if each H ∈ H has property A, then G has property A. This result leads to new classes of groups that hav...

Property a as Metric Amenability and Its Applications to Geometry

A Liouville Property of Holomorphic Maps

We prove a Liouville property of holomorphic maps from a complete Kähler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kähler manifold with a certain assumption on the sectional curvature.

On the Liouville Property for Divergence Form Operators

In this paper we construct a bounded strictly positive function õ such that the Liouville property fails for the divergence form operator L ≥ r(õ2r). Since in addition ∆õÛõ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schrödinger operators.