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6. Hyperfiniteness and related notions

Let $X$ be a Polish space.


    1. Add remark

      Problem 6.1.

      Let $f_n : X \rightarrow [0,1]$ Borel for $n \in \omega$. Let $I \subset {Borel}(X)$ a $\sigma$-ideal. Let $F_n = \frac{1}{n} \sum_{m < n} f_m$. Does $(F_n)_{n \in \omega}$ admit a subsequence which converges outside a set in $I$?
          We are especially interested in the case where $I$ is the ideal of Ramsey null sets. The answer is yes if $I$ is the ideal of $\mu$-null Borel sets for $\mu$ a Borel probability measure on $X$. The answer is no if $I$ is the ideal of meager Borel sets.


        • Add remark

          Problem 6.2.

          Is every orbit equivalence relation induced by a Borel action of the Grigorchuk group hyperfinite?

            •     The following seems closely related to the union problem. Perhaps someone can elaborate on whether there is a direct relationship.

              Add remark

              Problem 6.3.

              Let $E$ be a CBER on $X$. Suppose there is a Borel binary relation $<$ on $X$ such that the restriction of $<$ to each $E$-class is a linear order isomorphic to $\Z^2$ with the lexicographic order (“A $\Z$-line of $\Z$-lines”). Is $E$ hyperfinite?

                •     Call a group $\Gamma$ strongly hyperfinite if every orbit equivalent relation $E$ of a free*** continuous action of $\Gamma$ on a compact, 0-dim. Polish space admits a continuous reduction to $\mathbb{E}_0$.

                  *** There seemed to be some uncertainty about including free-ness.

                  Add remark

                  Problem 6.4.

                  1. Which groups are strongly hyperfinite? ($\Z^d$ is for $d \in \omega$.)
                  2. Is strong hyperfiniteness equivalent to the following? : All $E$ as above are topologically orbit equivalent to an action of $\Z$.


                    • Add remark

                      Problem 6.5.

                      Is an increasing countable union of hyperfinite CBERs hyperfinite? What if we add some restrictions, e.g. involving Borel asymptotic dimension?


                        • Add remark

                          Problem 6.6.

                          Is a Borel graph with finite asi (asymptotic separation index) hyperfinite? Does such a graph always have asi $\leq 1$?

                              Cite this as: AimPL: Descriptive graph theory, available at http://aimpl.org/descriptgraph.