Thompson群V的子群的图论刻画
A graph-theoretical characterisation of subgroups of Thompson's group $V$
AI总结：
该研究给出Thompson群V有限生成子群的图论刻画，据此证明相关群嵌入性结论，为Lehnert猜想提供证据，并确定部分群无法嵌入V。
AI中文摘要：
我们对Thompson群V的有限生成子群给出了图论刻画：一个有限生成群可嵌入V当且仅当它具有忠实的上下文无关作用，或等价地属于由Matucci与后三位作者新近引入的上下文无关图的转移群类CF-TR。利用该刻画，我们在多个方向证明了结果：- 所有已知具有余上下文无关字问题的群均可嵌入V，为Lehnert猜想提供了证据。- V的每个有限生成子群要么是几乎阿贝尔的，要么包含一个非阿贝尔自由半群，由此可得中间增长群无法嵌入Thompson群V。我们进一步研究由互为极限或覆盖的图定义的转移群之间的关系，并证明多项式增长上下文无关图的转移群的性质。最后，我们证明Basilica群和Hanoï Towers群无法嵌入V，该结论利用了这些群及Thompson群V的自然作用的Schreier图的几何性质。
英文摘要：
We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$: a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in $V$, providing evidence towards Lehnert's conjecture. - Each finitely generated subgroup of $V$ is either virtually abelian, or contains a free non-abelian semigroup. It follows that groups of intermediate growth do not embed in Thompson's $V$. We further study the relation between transition groups defined by graphs that are limits or covers of each others, and prove properties of transition groups of context-free graphs of polynomial growth. Finally, we prove that the Basilica and Hanoï Towers groups do not embed in $V$. This uses the geometry of Schreier graphs of the natural actions of these groups and of Thompson's $V$.
