For a fixed countably infinite structure Γ with finite relational signature τ , we study the following computational problem: input are quantifier-free τ-universal quantification are forbidden). We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure ∆ with a finite relational… (More)
We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are p-divisible for infinitely many primes p, or groups of the form L p∈I Z(p ∞), where I is an infinite set of primes.
In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on M X , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ… (More)
Consider a standard probability space (X, µ), i.e., a space isomorphic to the unit interval with Lebesgue measure. We denote by Aut(X, µ) the au-tomorphism group of (X, µ), i.e., the group of all Borel automorphisms of X which preserve µ (where two such automorphisms are identified if they are equal µ-a.e.). A Borel equivalence relation E ⊆ X 2 is called… (More)
Consider a measure-preserving action Γ (X, µ) of a countable group Γ and a measurable cocycle α : X × Γ → Aut(Y) with countable image , where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ X is non-amenable, then there does not exist a countable-to-one Borel… (More)