Strongly approximately transitive group actions, the Choquet-Deny theorem, and polynomial growth

Strongly Approximately Transitive Group Actions, the Choquet-deny Theorem, and Polynomial Growth

Transitive Group Actions

Every action of a group on a set decomposes the set into orbits. The group acts on each of the orbits and an orbit does not have sub-orbits (unequal orbits are disjoint), so the decomposition of a set into orbits could be considered as a “factorization” of the set into “irreducible” pieces for the group action. Our focus here is on these irreducible parts, namely group actions with a single orbit.

Cylindric Billiards and Transitive Lie Group Actions

A conjecture is formulated and discussed which provides necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup G of the orthogonal group O(d) (d being the dimension of the billiard in question) be transitive on the unit sphere Sd−1. If C1, . . . , Ck ar...

Group actions and Helly’s theorem

We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly’s 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encoded in a property that we introduce called “property FAr”, which reduces to Serre’s property FA when r = 1. T...

Generically Multiple Transitive Algebraic Group Actions

With every nontrivial connected algebraic group G we associate a positive integer gtd(G) called the generic transitivity degree of G and equal to the maximal n such that there is a nontrivial action of G on an irreducible algebraic variety X for which the diagonal action of G on Xn admits an open orbit. We show that gtd(G) 6 2 (respectively, gtd(G) = 1) for all solvable (respectively, nilpotent...