关于一个扭曲的辛格猜想
On a twisted Singer conjecture
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中文总结 AI 辅助
研究辛格猜想的自然类似物，即扭曲\(L^2\)-贝蒂数在下中间维度外消失的情况，其不变量与无限循环覆盖有关，还与作者之前猜想相关，且对一类闭算术双曲流形证明了该猜想。
中文摘要 AI 辅助
根据辛格猜想，闭非球面流形的\(L^2\)-贝蒂数预计在中间维度之外消失。本文研究了辛格猜想的一个自然类似物，即关于扭曲\(L^2\)-贝蒂数在下中间维度之外消失的情况。这些不变量可被视为无限循环覆盖的\(L^2\)-贝蒂数，取决于第一上同调类。这与作者之前的一个猜想相关，该猜想将扭曲\(L^2\)-欧拉特征与瑟斯顿范数联系起来，并且我们对一类闭算术双曲流形证明了该猜想。
英文摘要
According to the Singer conjecture, the $L^2$-Betti numbers of a closed aspherical manifold are expected to vanish outside the middle dimension. In this paper, we study a natural analog of the Singer conjecture concerning the vanishing of twisted $L^2$-Betti numbers outside the lower middle dimension. These invariants can be thought of as $L^2$-Betti numbers of an infinite cyclic cover, depending on a first cohomology class. This is related to a previous conjecture by the author, which connects the twisted $L^2$-Euler characteristic to the Thurston norm, and which we prove for a class of closed arithmetic hyperbolic manifolds.
