High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on non-trivial algebraic techniques. In particular, no random or combinatorial construction of bounded degree HDXs is known. As a result, our understanding of these objects is limited. The degree of an $i$-face in an HDX is the number of $(i+1)$-faces containing it. In this work we construct HDXs whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular $k$-dimensional HDX $X$ and outputs another $k$-dimensional HDX $\widehat{X}$ with twice as many vertices. While the degree of vertices in $\widehat{X}$ grows, the degree of the $(k-1)$-faces in $\widehat{X}$ stays the same. As a result, we obtain a new `algebra-free' construction of HDXs whose $(k-1)$-face degree is bounded. Our algorithm is based on a simple and natural generalization of the construction by Bilu and Linial (Combinatorica, 2006), which build expanders using lifts coming from edge signings. Our construction is based on local lifts of HDXs, where a local lift is a complex whose top-level links are lifts of links in the original complex. We demonstrate that a local lift of an HDX is an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. We use this technique to construct bounded degree high dimensional expanders with links that have arbitrarily large diameters.

翻译：高维扩展器（HDXs）是扩展器图在超图上的推广。由于其广泛的应用，它们在数学和理论计算机科学领域得到了深入研究。与扩展器图类似，当高维扩展器具有有界度性质（即每个顶点关联的边数有界）时，对应用尤其重要。然而，目前已知具有此性质的构造寥寥无几，且均依赖于非平凡的代数技术。特别是，尚不存在有界度高维扩展器的随机或组合构造。因此，我们对这类对象的理解仍然有限。在高维扩展器中，一个$i$-面的度是指包含它的$(i+1)$-面的数量。本文中，我们构造了高维面具有有界度的高维扩展器。具体而言，我们提出了一种基本且确定性的算法：该算法以正则$k$维高维扩展器$X$作为输入，并输出另一个顶点数翻倍的$k$维高维扩展器$\widehat{X}$。虽然$\widehat{X}$中顶点的度会增长，但其$(k-1)$-面的度保持不变。由此，我们获得了一种新的“无代数”构造方法，能够生成$(k-1)$-面度有界的高维扩展器。我们的算法基于Bilu和Linial（Combinatorica, 2006）构造的自然推广，该工作利用边符号标记产生的提升来构建扩展器。我们的构造基于高维扩展器的局部提升，其中局部提升是一种复形，其顶层链环是原始复形中链环的提升。我们证明了在多数情况下，高维扩展器的局部提升仍然是高维扩展器。此外，将局部提升与现有有界度构造相结合，可以生成新的有界度高维扩展器族，其链环特性与以往构造有显著差异。我们利用该技术构造了链环直径可任意大的有界度高维扩展器。