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 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. For every large enough $D$, we use this technique to construct families of bounded degree HDXs with links that have diameter $\geq D$.

翻译：高维扩展器（HDXs）是扩展图在超图上的推广。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。与扩展图类似，有界度的高维扩展器（即每个顶点相邻的边数有界）在应用中尤其重要。然而，目前已知具有此性质的构造寥寥无几，且均依赖于代数技术。特别是，尚不存在随机或组合的有界度高维扩展器构造。因此，我们对此类对象的理解仍然有限。在高维扩展器中，$i$-维面的度是指包含它的$(i+1)$-维面的数量。本文构造了高维面具有有界度的高维扩展器。具体方法是提出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，输出另一个具有两倍顶点数的$k$维高维扩展器$\widehat{X}$。虽然$\widehat{X}$中顶点的度会增长，但其$(k-1)$-维面的度保持不变。由此，我们获得了一种新的“无代数”高维扩展器构造，其$(k-1)$-维面度有界。我们的算法基于对Bilu和Linial（Combinatorica, 2006）构造的自然推广，他们利用边符号标记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，其中局部提升是指其顶层链环为原复形链环提升的复形。我们证明了在多数情况下，高维扩展器的局部提升仍为高维扩展器。此外，将局部提升与现有有界度构造相结合，可以生成具有与以往显著不同链环结构的新有界度高维扩展器族。对于任意足够大的$D$，我们利用该技术构造了链环直径$\geq D$的有界度高维扩展器族。