关联复形的实现与 profinite 刚性
Cubical residual complexes and profinite rigidity of real hyperbolic lattices
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中文总结 AI 辅助
本文通过可除填充与完备胞腔关联复形实现带标记 profinite 群，证明离散与完备作用的外自同构群一致，并应用于 Kleinian 群获得 PL 实现与边界子群对应。
中文摘要 AI 辅助
我们通过可除填充和完备胞腔关联复形证明了带标记 profinite 群的实现定理。对于闭曲面上的非分离曲线，可除曲线幂商实现割曲面子群的闭包作为精确交集，并确定完整的 profinite 割树。对于非分离填充对，完备对偶方形复形是两个割树乘积的闭交叉子复形。其关联映射实现给定的离散曲面作用，直至单个 profinite 平移。同样的论证适用于连通、局部有限、有限维正则 CW 复形上的自由余紧作用。图版本允许有限顶点稳定子，并仅假设在共尾多个特征有限商上具有等变关联同构。有限交集性质提供相容映射，且相同的构造识别离散和完备带标记作用的外自同构群。完全单纯形-面关联给出有限单纯复形的 PL 实现。我们将这些结果应用于 Kleinian 群。在格的情形，积分上同调比较、Massey 三元积和素数周期轨道的精确对应产生所需的曲面标记，而单位不变可除轨道填充处理尖点情形。对于任意有限生成 Kleinian 群，等变图标记给出边界子群对应和带标记紧核的 PL 实现。一个 Schottky 例子表明，在格环境之外，此类标记通常不能省略。
英文摘要
We study profinite rigidity of real hyperbolic lattices, with the main application in dimension three. The basic tool is a profinite version of the dual cube complex of a pair of filling curves on a surface, which we call a profinite cubical residue complex. We prove that an equivariant isomorphism between the completions of two such complexes is induced, after conjugation by a single element, by an isomorphism of the discrete groups and complexes, and that the same holds for free cocompact cellular actions of finitely generated residually finite groups on connected, locally finite, finite dimensional regular CW complexes. On a closed orientable surface of genus at least two, a theory of divisible fillings determines the completed cut tree of a nonseparating simple closed curve. Combined with the homological intersection criterion of Boggi and Zalesskii, this determines the completed dual square complex of a nonseparating filling pair, and hence realizes every isomorphism of profinite surface groups which preserves the two marked cyclic subgroups. In dimension three we combine these results with virtual fibering, Liu's profinite invariants of hyperbolic $3$-manifolds and the goodness of $3$-manifold groups. Massey products and virtual domination give cohomological integrality, and pseudo-Anosov dynamics, periodic orbit traces and cross sections of suspension flows give a correspondence of prime periodic orbits together with the fiber curves needed for the realization. Profinite rigidity of closed and of cusped lattices follows, the cusped case through cyclic orbifold fillings, and we obtain $\Out(Γ)\cong\Out(\whΓ)$ for every lattice of $\PSL_2(\C)$.
