论文标题
酰基叠双曲线组指数方程的溶液结构
Structure of solutions of exponential equations in acylindrically hyperbolic groups
论文作者
论文摘要
让$ g $是在双曲线空间上表现酰基的一组，让$ e $为$ g $的指数方程。我们表明，$ e $等同于成对独立方程的有限系统的有限析取，它们要么在几乎循环亚组或椭圆形上是圆锥形的。我们还获得了$ e $的解决方案集的描述。在$ g $相对于外围亚组集合$ \ {h_λ\} _ {λ\inλ} $的集合中，我们获得了更强的结果。特别是，在这种情况下，我们证明了$ g $上的指数方程的解决方案集为$ \ mathbb {z} $ - 半线性时，并且仅当每个$h_λ$，$h_λ$，$ phinλ$中的指数方程的解决方案集为$ \ mathbb {z} $ - {z} $ - 半emilinear。我们可以根据弱的前爆发算术中的可定义集，对指数方程的有限系统有限析出和对相对双曲线组的不等式的有限析出获得类似的结果。
Let $G$ be a group acting acylindrically on a hyperbolic space and let $E$ be an exponential equation over $G$. We show that $E$ is equivalent to a finite disjunction of finite systems of pairwise independent equations which are either loxodromic over virtually cyclic subgroups or elliptic. We also obtain a description of the solution set of $E$. We obtain stronger results in the case where $G$ is hyperbolic relative to a collection of peripheral subgroups $\{H_λ\}_{λ\in Λ}$. In particular, we prove in this case that the solution sets of exponential equations over $G$ are $\mathbb{Z}$-semilinear if and only if the solution sets of exponential equations over every $H_λ$, $λ\in Λ$, are $\mathbb{Z}$-semilinear. We obtain an analogous result for finite disjunctions of finite systems of exponential equations and inequations over relatively hyperbolic groups in terms of definable sets in the weak Presburger arithmetic.
