论文标题
虚拟理性的贝蒂数字数量
Virtual rational Betti numbers of nilpotent-by-abelian groups
论文作者
论文摘要
在本文中，我们研究了虚拟的betti数字数量的nilpotent-by-abelian $ g $，其中abelianization $ n/n'$的nilpotent part $ n $ $ n $可满足某些驯服财产。更准确地说，我们证明，如果$ n/n'$是$ 2（c（n-1）-1）$ - tame tame as a $ g/n $ module，$ c $ nilpotency类$ n $，则$ \ mathrm {vb} _j（g）：= \ sup_ {m \ in \ Mathcal {a} _g} \ dim_ \ dim_ \ dimbb {q} h_j（m，\ mathbb {q}）$是所有$ 0 \ leq j \ leq leq n $ a $ n $ qu \ n $ n $ n $ qu \ n $ n $ s n $ qu \ n $ n $ n $ n $ $ g $的有限指数子组。
In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N'$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N'$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $\mathrm{vb}_j(G):=\sup_{M\in\mathcal{A}_G}\dim_\mathbb{Q} H_j(M,\mathbb{Q})$ is finite for all $0\leq j\leq n$, where $\mathcal{A}_G$ is the set of all finite index subgroups of $G$.
