We consider the problem whether for a group $G$ there exists a constant $\mathrm{\Lambda }(G)>1$ such that for any $(r,s)$-matrix $A$ over the integral group ring $\mathbb{Z}G$ the Fuglede–Kadison determinant of the $G$-equivariant bounded operator $L^2{(G)}^r\to L^2{(G)}^s$ given by right multiplication with $A$ is either one or greater or equal to $\mathrm{\Lambda }(G)$. If $G$ is the infinite cyclic group and we consider only $r=s=1$, this is precisely Lehmer’s problem.

中文翻译：

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任意组的 Lehmer 问题

我们考虑这个问题是否对于一个群体$G$存在一个常数$\mathrm{\Lambda }(G)>\mathrm{1个}$这样对于任何$(r,秒)$-矩阵$\mathrm{一种}$在积分群环上$\mathbb{Z}G$的 Fuglede-Kadison 行列式$G$-等变有界算子${\mathrm{大号}}^{\mathrm{2个}}{(G)}^r\to {\mathrm{大号}}^{\mathrm{2个}}{(G)}^{秒}$由右乘法给出$\mathrm{一种}$是一或大于或等于$\mathrm{\Lambda }(G)$. 如果$G$是无限循环群，我们只考虑$r=秒=\mathrm{1个}$，这正是 Lehmer 的问题。