Papers by filippo calderoni
Journal of the London Mathematical Society
We analyze Euclidean spheres in higher dimensions and the corresponding orbit equivalence relatio... more We analyze Euclidean spheres in higher dimensions and the corresponding orbit equivalence relations induced by the group of rational rotations from the viewpoint of descriptive set theory. It turns out that such equivalence relations are not treeable in dimension greater than 2. Then we show that the rotation equivalence relation in dimension n ≥ 5 is not Borel reducible to the one in any lower dimension. Our methods combine a cocycle superrigidity result from the works of Furman and Ioana with the superrigidity theorem for S-arithmetic groups of Margulis. We also apply our techniques to give a geometric proof of the existence of uncountably many pairwise incomparable equivalence relations up to Borel reducibility.
We explore countable ordered Archimedean groups from the point of view of descriptive set theory.... more We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation induced by the natural action of $\mathrm{GL}_2(\mathbb{Q})$ on $\mathrm{Ar}(\mathbb{Q}^2)$ is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau. In particular, we show that its potential class is not $\boldsymbol{\Pi}^0_3$. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theor...
Bulletin of the London Mathematical Society, 2022
In this paper we study the Borel structure of the space of leftorderings LO(G) of a group G modul... more In this paper we study the Borel structure of the space of leftorderings LO(G) of a group G modulo the natural conjugacy action, and by using tools from descriptive set theory we find many examples of countable left-orderable groups such that the quotient space LO(G)/G is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of F 2 on LO(F 2) is universal, and leverage this result to provide many other examples of countable left-orderable groups G such that the natural G-action on LO(G) induces a universal countable Borel equivalence relation.
The Bulletin of Symbolic Logic
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Papers by filippo calderoni