论文标题
图形产品的可准性
Semistability of Graph Products
论文作者
论文摘要
图$γ$上的{\ it Graph product} $ g $是一个定义的组：对于每个顶点$ v $ of $γ$，都有相应的非平凡组$ g_v $。组$ g $是$ g_v $的免费产品的商人，$ g_v $ by换向关系$ [g_v，g_w] = 1 $，对于$γ$中的所有相邻$ v $和$ w $。如果某些（等同于任何）有限连接的cw-complex $ x $，$ \π_1（x）= g $，有限的组$ g $具有$ \ infty $}的{\ it ressable基本组为$ \ infty $}众所周知，有一个有限的基本组有限呈现的基本小组包含许多其他类别的小组，但这是一个40年的问题，即所有有限呈有限介绍的团体是否在$ \ infty $中都有可半固定的基本组。我们的主要定理是一个组合结果。 It states that if $G$ is a graph product on a finite graph $Γ$ and each vertex group is finitely presented, then $G$ has non-semistable fundamental group at $\infty$ if and only if there is a vertex $v$ of $Γ$ such that $G_v$ is not semistable, and the subgroup of $G$ generated by the vertex groups of vertices adjacent to $v$ is finite （等效地$ lk（v）$是一个完整的图形，每个顶点组$ lk（v）$都是有限的）。因此，如果一个人知道哪些$ g $的顶点组不可半度且哪些是有限的，那么$γ$的基础检查确定$ g $是否具有$ \ infty $的半度性基本组。
A {\it graph product} $G$ on a graph $Γ$ is a group defined as follows: For each vertex $v$ of $Γ$ there is a corresponding non-trivial group $G_v$. The group $G$ is the quotient of the free product of the $G_v$ by the commutation relations $[G_v,G_w]=1$ for all adjacent $v$ and $w$ in $Γ$. A finitely presented group $G$ has {\it semistable fundamental group at $\infty$} if for some (equivalently any) finite connected CW-complex $X$ with $π_1(X)=G$, the universal cover $\tilde X$ of $X$ has the property that any two proper rays in $\tilde X$ are properly homotopic. The class of finitely presented groups with semistable fundamental group at $\infty$ is known to contain many other classes of groups, but it is a 40 year old question as to whether or not all finitely presented groups have semistable fundamental group at $\infty$. Our main theorem is a combination result. It states that if $G$ is a graph product on a finite graph $Γ$ and each vertex group is finitely presented, then $G$ has non-semistable fundamental group at $\infty$ if and only if there is a vertex $v$ of $Γ$ such that $G_v$ is not semistable, and the subgroup of $G$ generated by the vertex groups of vertices adjacent to $v$ is finite (equivalently $lk(v)$ is a complete graph and each vertex group of $lk(v)$ is finite). Hence if one knows which vertex groups of $G$ are not semistable and which are finite, then an elementary inspection of $Γ$ determines whether or not $G$ has semistable fundamental group at $\infty$.
