Incomparable Actions of Free Groups
نویسندگان
چکیده
Suppose that X is a standard Borel space, E is a countable Borel equivalence relation on X, and μ is an E-invariant Borel probability measure on X. We consider the circumstances under which for every countable non-abelian free group Γ, there is a Borel sequence (·r)r∈R of free actions of Γ on X, generating subequivalence relations Er of E with respect to which μ is ergodic, with the further property that (Er)r∈R is an increasing sequence of relations which are pairwise incomparable under μ-reducibility. In particular, we show that if E satisfies a separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on X, generating a subequivalence relation of E with respect to which μ is ergodic.
منابع مشابه
Borel reductions of profinite actions of SLn(Z)
Greg Hjorth and Simon Thomas proved that the classification problem for torsion-free abelian groups of finite rank strictly increases in complexity with the rank. Subsequently, Thomas proved that the complexity of the classification problems for p-local torsion-free abelian groups of fixed rank n are pairwise incomparable as p varies. We prove that if 3 ≤ m < n and p, q are distinct primes, the...
متن کاملThe Classification of Torsion-free Abelian Groups of Finite Rank up to Isomorphism and up to Quasi-isomorphism
We prove that the isomorphism and quasi-isomorphism relations on the p-local torsion-free abelian groups of fixed finite rank n are incomparable with respect to Borel reducibility.
متن کاملTHE CLASSIFICATION PROBLEM FOR p-LOCAL TORSION-FREE ABELIAN GROUPS OF FINITE RANK
Let n ≥ 3. We prove that if p 6= q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank n are incomparable with respect to Borel reducibility.
متن کاملBernoulli Actions of Low Rank Lattices and Countable Borel Equivalence Relations
Using Zimmer’s cocycle superrigidity theorems [27], we obtain Borel nonreducibility results for orbit equivalence relations arising from Bernoulli actions of suitably chosen low-rank lattices in real and p-adic Lie groups. In particular, for p a prime, let Ep (respectively Fp) denote the orbit equivalence relation arising from any nontrivial Bernoulli action of PSL2(Z[ √ p ]) (respectively PSL2...
متن کاملTreeable equivalence relations
There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group – and hence, there are 20 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility
متن کامل