Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional (HD) expanders. We show that the naive approach to powering does not preserve HD expansion and define a new power operation, using geodesic walks on quotients of Bruhat–Tits buildings. Applying this operation results in HD expanders of higher degrees. The crux of the proof is a combinatorial study of flags of free modules over finite local rings. Their geometry describes links in the power complex, and showing that they are excellent expanders implies HD expansion for the power complex by Garland’s local-to-global technique. As an application, we use our power operation to obtain new efficient double samplers.

中文翻译：

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局部环上的自由标志和高维扩展器的供电

为扩展器图的邻接矩阵供电会产生更高程度的更好扩展器。在本文中，我们寻求高维 (HD) 扩展器的模拟操作。我们表明，简单的供电方法不能保留 HD 扩展并定义新的供电操作，使用 Bruhat-Tits 建筑物商上的测地线行走。应用此操作会导致更高程度的 HD 扩展器。证明的关键是对有限局部环上的自由模块标志的组合研究。它们的几何结构描述了电力复合体中的链接，并且表明它们是优秀的扩展器意味着通过 Garland 的本地到全局技术对电力复合体进行 HD 扩展。作为一个应用程序，我们使用我们的幂运算来获得新的高效双采样器。