论文标题
fuglede-kadison的决定因素和莱默的常数
Fuglede-Kadison determinants over free groups and Lehmer's constants
论文作者
论文摘要
莱默（Lehmer）的著名问题询问了具有整数系数的多项式措施的一组措施是否在2019年的差距中，吕克（Lück）将这个问题扩展到了一般组的fuglede-kadison决定因素，他定义了该小组的莱默（Lehmer）常数以衡量这种差距。 在本文中，我们计算了非循环无环组的fuglede-kadison决定因素的新值，这产生了新的上限$ \ frac {2} {\ sqrt {\ sqrt {3}} $，用于Lehmer所有具有非循环免费子组的Torsion torsion torsion constants。 我们的证明使用了fuglede-kadison的决定因素与在Cayley图上的随机步行之间的关系，以及Bartholdi和Dasbach-Lalin的作品。 此外，通过$ l^2 $分数的胶合公式，我们表明，莱默的无限数量的柔毛3个manifolds组的常数以上甚至小于$ \ frac {2} {\ sqrt {3}} $的值。
Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, Lück extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer's constants of the group to measure such a gap. In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound $\frac{2}{\sqrt{3}}$ for Lehmer's constants of all torsion-free groups which have non-cyclic free subgroups. Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of Bartholdi and Dasbach-Lalin. Furthermore, via the gluing formula for $L^2$-torsions, we show that the Lehmer's constants of an infinite number of fundamental groups of hyperbolic 3-manifolds are bounded above by even smaller values than $\frac{2}{\sqrt{3}}$.
