Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families

On equationally Noetherian and residually finite groups

The aim of this paper is to compare and contrast the class of residually finite groups with the class of equationally Noetherian groups - groups over which every system of coefficient-free equations is equivalent to a finite subsystem. It is easy to construct groups that are residually finite but not equationally Noetherian (e.g. the direct sum of all finite groups) or vice versa (e.g. the additive group $(\mathbb{Q},+)$ of the rationals). However, no explicit examples that are finitely generated seem to appear in the literature. In this paper, we show that among finitely generated groups, the…

自由-循环群是方程可约的

Free-by-cyclic groups are equationally Noetherian

本文证明自由-循环群是方程可约的，并展示了其指数增长速率集合的良序性，同时揭示了相关群的非初等作用和分裂性质。

强简洁性与方程诺特群

Strong conciseness and equationally Noetherian groups

本文研究了群中字的简洁性、强简洁性与方程诺特群的关系，证明了特定完备群中|w{G}|<2^ℵ₀时w{G}有限，得出多个群类中每个字均强简洁的结论，扩展了已知简洁性性质。

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…

柱形双曲群中的增长率

The rates of growth in an acylindrically hyperbolic group

本文研究柱形双曲群的增长率问题，在有限生成集的M次幂含双曲元素、群为方程诺特的条件下，证明其增长率集合良序，推广了双曲群的已知结果，可应用于秩1单李群格、相对双曲群及一类3-流形基本群。

可着色分层双曲群的方程 Noether 性

On Equational Noetherianity of Colorable Hierarchically Hyperbolic Groups

本文考察固定有限生成群到严格非柱状可着色分层双曲群的同态，证明了这类群均为方程 Noether 群。