Finding non-trivial elements and splittings in groups
Maurice Chiodo
备注：
16 pages, revised version, minor corrections made, additional results on algorithms to construct embeddings and enumerate subgroups. v3: minor changes to typesetting and theorem numberings, minor corrections made
发表信息：
M. Chiodo, `Finding non-trivial elements and splittings in groups', J. Algebra. 331, 271-284 (2011)
It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: `Is every finitely generated perfect group the normal closure of one element?' We prove a weakened version of our question: there is no general procedure to pick a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose a finite presentation of a non-trivial free product into two non-trivial finitely presented factors, nor one to construct an embedding from one finitely presented group into another in which it embeds. We apply our results to show that a construction by Stallings on splitting groups with more than one end can never be made algorithmic, nor can the process of splitting connect sums of non-simply connected closed 4-manifolds.
Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block
在本文中，我们证明了以下结果。设$G$是特征为$p\geq 0$的代数闭域$F$上的例外Lie型单连通线性代数群，$u\in G$是非恒等幂幺元.令$ϕ$是$G$的非平凡不可约表示。则$ϕ(u)$的Jordan标准形至多包含一个非平凡块当且仅当$G$是$G_2$型的，$u$是正则幂幺元，且$\dim ϕ\leq 7$ .注意，若一个非平凡的幂幺元由一个若当型有一个非平凡块的矩阵表示，则简单经典代数群的不可约表示由I. D. Suprunenko（Unipotent elements of non-prime order in the representations of the classical代数群群：two big Jordan blocks，J. Math. Sci. 199（2014），350 - 374.
Splittings and automorphisms of relatively hyperbolic groups
研究了相对双曲群G的自同构。当G是单端的，我们描述出（G）使用首选的JSJ树的子群，几乎是循环或抛物线。特别地，当G是相对双曲的时，Out（G）实际上是由GL_n（Z）的映射类群和子群固定某些基元素构成的。当允许更一般的抛物群时，GL_n（Z）的这些子群必须用McCool群代替：抛物群的自同构平凡地（即通过共轭）作用于某些子群。给定一个双曲群G的一个反常拟凸子群P，我们认为G是相对于P的双曲群，我们应用前面的分析来描述P的自同构群Out（P to G）延伸到G：它实际上是一个McCool群。如果Out（P to G）是无限的，则P是G的一个分裂中的一个顶点群.如果P是无挠的，那么Out（P to G）是VF类型的，特别是VF呈现的。我们还确定了当G是相对双曲的时，Out（G）是无穷大的。有趣的情况是当G是无穷终结的并且有挠。当G是双曲群时，我们证明了Out（G）是无穷的当且仅当G在一个具有无穷中心的极大虚循环子群上分裂.在一般情况下，我们表明，无穷的出（G）来自于存在一个分裂与无限多的扭曲，或有一个顶点群，这是极大的抛物与无穷多个自同构平凡的事件棱道群。
