Generalized torsion for hyperbolic 3‐manifold groups with arbitrary large rank
Generalized torsion for hyperbolic 3‐manifold groups with arbitrary large rank
复制标题
具有任意大秩的双曲 3 流形群的广义扭转
DOI：
10.1112/blms.12784
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发表时间：
2022
影响因子：
0.9
通讯作者：
Teragaito Masakazu
中科院分区：
文献类型：
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作者：
Ito Tetsuya;Motegi Kimihiko;Teragaito Masakazu
设G$G$是一个群，g$g$是G$G$中的一个非平凡元素。如果g$g$的共轭的某个非空有限积等于平凡元，则g$g$称为广义扭转元。据我们所知，我们没有双曲三维流形群的广义扭转元的秩显式已知大于2。本文的目的是证明对于给定的整数n>1$n > 1$，有无穷多个闭双曲三维流形Mn$M_n$，它们具有以下性质：（i）Mn$M_n$的Heegaard亏格是n$n$，（ii）π1（Mn）$\pi _1（M_n）$的秩是n$n$，（ii）π1（Mn）$\pi _1（M_n）$有广义挠元。此外，我们还可以选择Mn$M_n$作为同调透镜空间，使得广义挠元的阶数是任意大的.
Let G$G$ be a group and g$g$ a non‐trivial element in G$G$. If some non‐empty finite product of conjugates of g$g$ equals to the trivial element, then g$g$ is called ageneralized torsion element. To the best of our knowledge, we have no hyperbolic 3‐manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer n>1$n > 1$ there are infinitely many closed hyperbolic 3‐manifolds Mn$M_n$ which enjoy the property: (i) the Heegaard genus of Mn$M_n$ is n$n$, (ii) the rank of π1(Mn)$\pi _1(M_n)$ is n$n$, and (ii) π1(Mn)$\pi _1(M_n)$ has a generalized torsion element. Furthermore, we may choose Mn$M_n$ as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.