论文标题
正式的解决方案和酰基叠双曲线的一阶理论
Formal solutions and the first-order theory of acylindrically hyperbolic groups
论文作者
论文摘要
我们将Merzlyakov关于非亚伯自由群体的一阶理论的定理概括为所有酰基双曲线群。作为推论，我们推断出，如果$ g $是一个酰基透明的组，$ e（g）$表示$ g $的独特最大有限正常亚组，则表示$ g $，$ g $和hnn扩展$ g \ dot {\ ast} _ {e（g）} $ g \ e a aS $很琐碎，具有相同的$ \ forall \存在$ - 理论。结果，我们证明了由Casals-Ruiz，Garreta和de LaNuezGonzález提出的以下猜想：酰基杂质的双曲基团具有微不足道的阳性理论。特别是，人们恢复了Bestvina，Bromberg和Fujiwara证明的结果，并指出，只有明显的例外，酰基叠纤维双曲线组的语言亚组具有无限的宽度。
We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez González: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.
