论文标题
接近上升的hnn扩展和无限性的共同性的组合结果
Near Ascending HNN-Extensions and a Combination Result for Semistability at Infinity
论文作者
论文摘要
Infinity的半可见性是有限呈现的组的渐近性质，以有效地定义了1端组的无穷大基本组。无论是否有限呈现的群体在Infinity中都有可分类的基本群体，这是一个开放的问题。虽然已知许多类别仅包含在无穷大组中的半含量，但此类组只有少数组合结果。我们的主要定理就是这样的结果。 主要定理。假设$ g $是连接的减少组的基本组，其中每个边组是无限且有限生成的，并且每个顶点组有限地呈现，并且在Infinity处有1端且可在Infinity中半固定，或者具有一组有限索引。然后，$ g $是1端，在无穷大。 该结果证明的一个重要部分是以下内容的可准性部分： 定理。假设$ h_0 $是一个有限限制的组，$ h_1 $是$ h_0 $，$ ϕ：h_1 \ to h_0 $的有限索引子组，是单态性，$ g = h_0 \ ast_2 $是由此产生的HNN扩展。然后，$ g $是1端，在无穷大。如果另外，$ h_0 $是1端，那么$ g $仅在无限范围内仅连接。
Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely presented groups have semistable fundamental group at infinity. While many classes of groups are known to contain only semistable at infinity groups, there are only a few combination results for such groups. Our main theorem is such a result. Main Theorem. Suppose $G$ is the fundamental group of a connected reduced graph of groups, where each edge group is infinite and finitely generated, and each vertex group is finitely presented and either 1-ended and semistable at infinity or has an edge group of finite index. Then $G$ is 1-ended and semistable at infinity. An important part of the proof of this result is the semistability part of the following: Theorem. Suppose $H_0$ is an infinite finitely presented group, $H_1$ is a subgroup of finite index in $H_0$, $ϕ:H_1\to H_0$ is a monomorphism and $G=H_0\ast_ϕ$ is the resulting HNN extension. Then $G$ is 1-ended and semistable at infinity. If additionally, $H_0$ is 1-ended, then $G$ is simply connected at infinity.
