arXiv subjects

Robin D. Tucker-Drob

Publications and source records attributed to Robin D. Tucker-Drob.

4 recordsLinked to original sources

arXiv2026

One-ended spanning subforests and treeability of groups

We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its closed subgroups. This provides the first examples of one-ended nonamenable groups which are measure strongly treeable. In higher dimensions, we also prove a dichotomy that the fundamental group of a closed aspherical 3-manifold is either amenable or has strong ergodic dimension 2. Our main technical tool is a method for finding measurable treeings of Borel planar graphs by constructing one-ended spanning subforests in their planar dual. Our techniques for constructing one-ended spanning subforests also give a complete classification of the locally finite pmp graphs which admit Borel a.e. one-ended spanning subforests.

math.GR↗

arXiv2014

Borel structurability on the 2-shift of a countable group

We show that for any infinite countable group $G$ and for any free Borel action $G \curvearrowright X$ there exists an equivariant class-bijective Borel map from $X$ to the free part $\mathrm{Free}(2^G)$ of the $2$-shift $G \curvearrowright 2^G$. This implies that any Borel structurability which holds for the equivalence relation generated by $G \curvearrowright \mathrm{Free}(2^G)$ must hold a fortiori for all equivalence relations coming from free Borel actions of $G$. A related consequence is that the Borel chromatic number of $\mathrm{Free}(2^G)$ is the maximum among Borel chromatic numbers of free actions of $G$. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic $G$-equivariant map to $2^G$ lands in the free part. As a corollary we obtain that for every $ε> 0$, every free pmp action of $G$ has a free factor which admits a $2$-piece generating partition with Shannon entropy less than $ε$. This generalizes a result of Danilenko and Park.

math.DS↗

arXiv2012

Shift-minimal groups, fixed price 1, and the unique trace property

A countable group Γis called shift-minimal if every non-trivial measure preserving action of Γweakly contained in the Bernoulli shift of Γon ([0,1]^Γ,λ^Γ) is free. We show that any group Γwhose reduced C^*-algebra admits a unique tracial state is shift-minimal, and that any group Γadmitting a free measure preserving action of cost>1 contains a finite normal subgroup N such that Γ/N is shift-minimal. Any shift-minimal group in turn is shown to have trivial amenable radical. Recurrence arguments are used in studying invariant random subgroups of a wide variety of shift-minimal groups. We also examine continuity properties of cost in the context of infinitely generated groups and equivalence relations. A number of open questions are discussed which concern cost, shift-minimality, C^*-simplicity, and uniqueness of tracial state on C^*_r(Γ).

math.GR↗

arXiv2012

Weak equivalence and non-classifiability of measure preserving actions

Abért-Weiss have shown that the Bernoulli shift s of a countably infinite group Γis weakly contained in any free measure preserving action (mpa) b of Γ. We establish a strong version of this result, conjectured by Ioana, by showing that s \times b is weakly equivalent to b. This is generalized to non-free mpa's using random Bernoulli shifts. The result for free mpa's is used to show that isomorphism on the weak equivalence class of a free mpa does not admit classification by countable structures. This provides a negative answer to a question of Abért and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. An infinite residually finite group Γis said to have EMD if the action p of Γon its profinite completion weakly contains all ergodic mpa's of Γ, and Γis said to have property MD if i \times p weakly contains all mpa's of Γ, where i denotes the trivial action on a standard non-atomic probability space. Kechris asks if these two properties equivalent and we provide a positive answer by studying the relationship between convexity and weak containment.

math.DS↗