论文标题
相对DEHN FUCTION，度上嵌入的亚组和组合定理
Relative Dehn fuctions, hyperbolically embedded subgroups and combination theorems
论文作者
论文摘要
考虑以下一类，由一个组和有限的亚组集合组成： $ \ MATHCAL {C} = \ left \ {（g，\ \ \ \ \ \ \ Mathcal H）\ Mid \ text {$ \ Mathcal {h} $被倍增为$ G $} \ $} \ right \} $，并且 $ \ MATHCAL {d} = \ left \ {（g，\ \ \ \ \ \ \ mathcal h）\ mid \ text {$（g，\ mathcal h）$的相对dehn函数是很好的定义} \ right \}。$ $ 让$ g $是一个组，将每个顶点组$ g_v $分配为有限的组，分配了一个有限的子组$ \ mathcal {h} _v $的集合，每个边缘组$ g_e $ conjugate conjugate conjugate conjugate conjugate conjugate conjugate conjugate in \ mathcal in \ mathcal {hh} $ is $ if $ if $ if $ if $然后是$ g $的子组的有限集合$ \ MATHCAL {H} $，这样： $ \ bullet $如果每个$（g_v，\ mathcal {h} _v）$在$ \ mathcal c $中，则$（g，\ nathcal {h}）$在$ \ mathcal c $中。 $ \ bullet $如果每个$（g_v，\ mathcal {h} _v）$在$ \ mathcal d $中，则$（g，\ nathcal {h}）$在$ \ mathcal d $中。 $ \ bullet $对于任何顶点$ v $，对于g_v $中的任何$ g \，元素$ g $与某个$ q \ in \ mathcal {h} _v $中的元素相连，并且仅当$ g $ in \ in \ nath \ nathcal {h h h} $ in $ h \ in \ g $ conde of and \ mathcal {h} _v $。 该边缘组不被认为是有限生成的，并且它们不一定属于相邻顶点的亚组的外围集合是这项工作与文献中先前结果之间的主要区别。证明方法提供了相对DEHN函数的下限和上限，该函数根据顶点组的相对DEHN函数。这些边界在文献中概括并改善了类似的结果。
Consider the following classes of pairs consisting of a group and a finite collection of subgroups: $\mathcal{C}= \left\{ (G,\mathcal H) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right\}$ and $ \mathcal{D}= \left\{ (G,\mathcal H) \mid \text{the relative Dehn function of $(G,\mathcal H)$ is well-defined} \right\}.$ Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a finite collection of subgroups $\mathcal{H}$ of $G$ such that: $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$, then $(G,\mathcal{H})$ is in $\mathcal C$. $\bullet$ If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$, then $(G,\mathcal{H})$ is in $\mathcal D$. $\bullet$ For any vertex $v$ and for any $g\in G_v$, the element $g$ is conjugate to an element in some $Q\in\mathcal{H}_v$ if and only if $g$ is conjugate to an element in some $H\in\mathcal{H}$. That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.
