We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

中文翻译：

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算术群的有限不变量

我们证明了具有同余子群性质的算术群的欧拉特征的符号是由有限补全决定的。相反，我们构建的例子表明这对于欧拉特征本身是不正确的，并且欧拉特征的符号在一般剩余有限类型群中不是有限的F. 我们的方法意味着类似的结果 $\ell^2$ -torsion 以及 Novikov-Shubin 不变量的强有限性陈述。