The Grushko decomposition of a finite graph of finite rank free groups: an algorithm

顶点链接与Grushko分解

Vertex links and the Grushko decomposition

本文提出一种多项式时间复杂度算法，通过分析CAT(0)方复形的顶点链接，构造具有循环边群的自由群图基本群的Grushko分解，并将一端CAT(0)方复形转化为满足强连通条件的复形。

Effective JSJ Decompositions

In this paper we describe an elimination process which is a deterministic rewriting procedure that on each elementary step transforms one system of equations over free groups into a finitely many new ones. Infinite branches of this process correspond to cyclic splittings of the coordinate group of the initial system of equations. This allows us to construct algorithmically Grushko's decompositions of finitely generated fully residually free groups and cyclic [abelian] JSJ decompositions of freely indecomposable finitely generated fully residually free groups. We apply these results to obtain a…

A decomposition theorem for higher rank Coxeter groups

In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups.

Automorphisms of Groups and a Higher Rank JSJ Decomposition I: RAAGs and a Higher Rank Makanin-Razborov Diagram

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct w…

群的自同构与高秩JSJ分解 II：单端情形

Automorphisms of Groups and a Higher Rank JSJ Decomposition II: The single ended case

本文研究满足弱无柱条件的层级双曲群(HHG)的自同构，通过构造高秩JSJ分解来编码单个自同构的动力学及HHG外自同构群的结构。

A Jordan decomposition for groups of finite Morley rank

We prove a Jordan decomposition theorem for minimal connected simple groups of finite Morley rank with non-trivial Weyl group. From this, we deduce a precise structural description of Borel subgroups of this family of simple groups. Along the way we prove a Tetrachotomy theorem that classifies minimal connected simple groups. Some of the techniques that we develop help us obtain a simpler proof of a theorem of Burdges, Cherlin and Jaligot.