论文标题
伪造的投射飞机的代数基本群体
Algebraic fundamental groups of fake projective planes
论文作者
论文摘要
基本的假射击平面组属于五十个不同的同构类别，每个复杂的共轭对。我们证明，对于他们的代数基本群体，情况并非如此：只有46个同构类别。我们表明，有四对复杂的偶联对伪造的射击平面是$ \ mathrm {aut}（\ mathbb {c}）$ - 等效，因此具有相互同构的代数基本组。所有其他代数基本组都通过明确的有限étale覆盖物显示出不同的代数基本组。作为副产品，这提供了第一个具有同构型涂鸦完成的半半密布谎言组中可相称但非同构晶格的示例。
Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are $\mathrm{Aut}(\mathbb{C})$-equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.
