剩余有限群的非 sof ic 圈积
Nonsofic wreath products of residually finite groups
AI总结：
本研究基于 OpenAI 发现首个非 sof ic 群的成果，推导得出满足特定条件的广义圈积为非 sof ic 的结论，该结论适用于多项式环等上的初等群对。
AI中文摘要：
本研究基于 OpenAI 发现首个非 sof ic 群的突破性成果展开，分析其底层证明机制并拓展应用。设 Γ<G 满足：集合 {g∈G | gΓg⁻¹≤Γ} 生成群 G，且 Γ 与 G 均具有性质 (T)；若 Γ 非正规，则广义圈积 (⊕_{G/Γ} ℤ/2ℤ) ⋊ G 为非 sof ic。上述假设对多项式环和 Laurent 多项式环上的初等群显式对成立，其中两群均为剩余有限且具有 Kazhdan 性质。
英文摘要：
This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $Γ<G$ be such that $\{g\in G:gΓg^{-1}\leqΓ\}$ generates $G$ as a group, and suppose that both $Γ$ and $G$ have property $(T)$. If $Γ$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/Γ}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_Γ G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.
