论文标题
块映射班级组及其有限属性
Block mapping class groups and their finiteness properties
论文作者
论文摘要
Cantor Surface $ \ MATHCAL C_D $是通过固定紧凑的表面$ y^d $（一个块）粘合副本，具有$ d+1 $边界组件，以树状的方式获得。对于固定的子组$ h <map（y^d）$，我们考虑子组$ \ mathfrak b_d（h）<map（\ mathcal c_d）$，其元素最终将块发送到块并像$ h $的元素一样；我们将$ \ mathfrak b_d（h）$称为块映射类组，其本地操作由$ h $规定。如此获得的组家族包含\ cite {sw21a，abf+21，fk04}的渐近映射类群。此外，对Farley的对称对称的汤普森群体有自然的陈述 - hughes \ cite {fh15}；特别是，它们为\ cite [问题5.37] {av20}提供了积极的答案。我们证明，当块为（孔）球或（孔）圆环时，$ \ mathfrak b_d（h）$是type $ f_n $的类型$ f_n $，并且仅当$ h $是类型$ f_n $的类型。结果，对于每个$ n $，$映射（C_D）$具有类型$ f_n $的子组，而不是$ f_ {n+1} $，其中包含$ \ Mathcal C_D $的每个紧凑型地下的映射类组。
A Cantor surface $\mathcal C_d$ is a non-compact surface obtained by gluing copies of a fixed compact surface $Y^d$ (a block), with $d+1$ boundary components, in a tree-like fashion. For a fixed subgroup $H<Map(Y^d)$ , we consider the subgroup $\mathfrak B_d(H)<Map(\mathcal C_d)$ whose elements eventually send blocks to blocks and act like an element of $H$; we refer to $\mathfrak B_d(H)$ as the block mapping class group with local action prescribed by $H$. The family of groups so obtained contains the asymptotic mapping class groups of \cite{SW21a,ABF+21, FK04}. Moreover, there is a natural surjection onto the family symmetric Thompson groups of Farley--Hughes \cite{FH15}; in particular, they provide a positive answer to \cite[Question 5.37]{AV20}. We prove that, when the block is a (holed) sphere or a (holed) torus, $\mathfrak B_d(H)$ is of type $F_n$ if and only if $H$ is of type $F_n$. As a consequence, for every $n$, $Map(C_d)$ has a subgroup of type $F_n$ but not $F_{n+1}$ which contains the mapping class group of every compact subsurface of $\mathcal C_d$.
