  The theory of $k$-regular graphs is closely related to group theory. Every
$k$-regular, bipartite graph is a Schreier graph with respect to some group
$G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The
goal of this paper is to begin to develop such a framework for $k$-regular
simplicial complexes of general dimension $d$. Our approach does not directly
generalize the concept of a Schreier graph, but still presents an extensive
family of $k$-regular simplicial complexes as quotients of one universal
object: the $k$-regular $d$-dimensional arboreal complex, which is itself a
simplicial complex originating in one specific group depending only on $d$ and
$k$. Along the way we answer a question from [PR12] on the spectral gap of
higher dimensional Laplacians and prove a high dimensional analogue of
Leighton's graph covering theorem. This approach also suggests a random model
for $k$-regular $d$-dimensional multicomplexes.
