  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 $\geq D$.
