arXiv 2512.045312025-12math.GRmath.DSmath.OA

一族连续统大小的非测度等价群

A continuum of non-measure equivalent groups

University of California San Diego（加州大学圣地亚哥分校）；University of Florida（佛罗里达大学）

Adrian Ioana, Robin Tucker-Drob

备注：: Added a new result showing that the ME-relation on the space of finitely generated groups is not smooth

AI总结：

该研究构造了一族连续统大小的可数群，它们两两非测度等价，具有性质(T)等。同时证明了有限生成群间测度等价关系$\simeq^{\mathrm{fg}}_{\mathrm{ME}}$是非光滑的，且在博雷尔可约性层次中高于可数博雷尔等价关系。

摘要：

我们构造了一族连续统大小的可数群$\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$，它们两两非测度等价，具有性质(T)（因此是有限生成的），所有阶的$\ell^2$-贝蒂数为零且无挠。我们还证明了有限生成群之间的测度等价关系$\simeq^{\mathrm{fg}}_{\mathrm{ME}}$是非光滑的，解决了S. Thomas的一个问题。此外，我们的证明表明$\simeq^{\mathrm{fg}}_{\mathrm{ME}}$在博雷尔可约性层次结构中高于每个可数博雷尔等价关系。

英文摘要：

We construct a continuum sized family $\{G_x\}_{x\in\{0,1\}^{\mathbb N}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero $\ell^2$-Betti numbers of all orders, and are torsion-free. We also prove that the equivalence relation $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ of measure equivalence between finitely generated groups is non-smooth, resolving a question of S. Thomas. Our proof moreover shows that $\simeq^{\mathrm{fg}}_{\mathrm{ME}}$ sits above every countable Borel equivalence relation in the Borel reducibility hierarchy.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。