Acylindrical hyperbolicity and existential closedness

Equations in acylindrically hyperbolic groups and verbal closedness

We describe solutions of the equation $x^ny^m=a^nb^m$ in acylindrically hyperbolic groups (AH-groups), where $a,b$ are non-commensurable special loxodromic elements and $n,m$ are integers with sufficiently large common divisor. Using this description and certain test words in AH-groups, we study the verbal closedness of AH-subgroups in groups. A subgroup $H$ of a group $G$ is called verbally closed if for any word $w(x_1,\dots, x_n)$ in variables $x_1,\dots,x_n$ and any element $h\in H$, the equation $w(x_1,\dots, x_n)=h$ has a solution in $G$ if and only if it has a solution in $H$. Main Theo…

群的图乘积中的轴双曲性与纯共轭自同构

Acylindrical Hyperbolicity and Pure Conjugating Automorphisms in Graph Products of Groups

该研究从两个视角刻画群的图乘积的轴双曲性，并确立纯共轭自同构的拓扑生成、闭性等性质，还给出阿贝尔群图乘积对应的典范拓扑半直积分解。