非正曲率下的模 $p$ 同调与挠同调增长
Mod $p$ and torsion homology growth in nonpositive curvature
AI总结：
本文通过计算直角 Artin 群剩余序列的模 p 同调增长并构造闭局部 CAT(0) 流形，证明了同调挠元可呈指数增长，从而揭示 Singer 猜想与 Lück 猜想至少有一个不成立。
摘要：
我们计算了直角 Artin 群的有限指数正规子群剩余序列的模 $p$ 同调增长。我们找到了其中与有理同调增长不同的例子，这意味着序列中子群的同调含有大量挠元。更精确地说，同调挠元随子群指数呈指数增长。对于奇素数 $p$，我们构造了在中间维数之外具有非零模 $p$ 同调增长的闭局部 CAT(0) 流形。这些例子表明，Singer 关于有理同调增长的猜想与 Lück 关于挠同调增长的猜想彼此不相容，因此其中至少有一个必定是错误的。
英文摘要：
We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and Lück's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong.
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