Decision problems in groups and extensions

组合群论是研究自由群以及如何根据生成元和关系将群表示为群的理论。有三个著名的问题，即字问题、共轭问题和同构问题，这三个问题最早由Dehn提出，并已被研究了一个多世纪。一般来说，解决这些问题是不可能的，但这些问题的良好解决方案往往在某些群体中自然而然地出现。在这些团体中，有自由团体和自由阿贝尔团体。最近几年，数学家们也研究了部分交换群--也称为直角Artin群(RAAG)--它们介于自由群和自由阿贝尔群之间。单词，共轭和同构问题在RAG中是已知的，但当它被移到几乎自由的RAG上时，几乎什么都不知道。在我的研究中，我的目标是通过研究共轭问题，结合使用代数、组合和几何方法，更多地了解虚拟自由RAG的自同构群。

Combinatorial group theory is the study of free groups and how groups can be given as a presentation based on generators and relations. There are three famous problems, namely the Word Problem, Conjugacy Problem and Isomorphism Problem, which were first posed by Dehn and have been studied for over a century. In general, it is not possible to solve these problems, but often nice solutions to these problems occur naturally in certain groups. Among such groups are free groups and free abelian groups. In more recent years, mathematicians have also studied partially commutative groups - also known as right angled Artin groups (RAAG) - which lie somewhere in between free groups and free abelian groups. The Word, Conjugacy and Isomorphism problems are known in RAAGs, but when this is moved to virtually free RAAGs, almost nothing is known. In my research I aim to understand more about the automorphism groups of virtually free RAAGs by working on the conjugacy problem, using a combination of algebraic, combinatorial and geometric methods.