We discuss the Heisenberg group $\Heis^n$ and its mappings from three perspectives.As a nilpotent Lie group, $\Heis^n$ can be viewed as a generalization of the real numbers, leading to new notions of base-$b$ expansions and continued fractions. As a metric space, $\Heis^n$ serves as an infinitesimal model (metric tangent space) of some sub-Riemannian manifolds and allows one to study derivatives of mappings between such spaces. As a subgroup of the isometry group of complex hyperbolic space $\Hyp^{n+1}_\C$, $\Heis^n$ becomes a large-scale model of a rank-one symmetric space and provides rigidity results in $\Hyp^{n+1}_\C$.After discussing homotheties and conformal mappings of $\Heis^n$, we show the convergence of base-$b$ and continued fraction expansions of points in $\Heis^n$, and discuss their dynamical properties. We then generalize to sub-Riemannian manifolds and their quasi-conformal and quasi-regular mappings. We show that sub-Riemannian lens spaces admit uniformly quasi-regular (UQR) self-mappings, and use Margulis--Mostow derivatives to construct for each UQR self-mapping of an equiregular sub-Riemannian manifold an invariant measurable conformal structure. Turning next to hyperbolic spaces, we recall the relationship between quasi-isometries of Gromov hyperbolic spaces and quasi-symmetries of their boundaries. We show that every quasi-symmetry of $\Heis^n$ lifts to a bi-Lipschitz mapping of $\Hyp^{n+1}_\C$, providing a rigidity result for quasi-isometries of $\Hyp^{n+1}_\C$. We conclude by showing that if $\Gamma$ is a lattice in the isometry group of a non-compact rank one symmetric space (except $\Hyp^1_\C = \Hyp^2_\R$), then every quasi-isometric embedding of $\Gamma$ into itself is, in fact, a quasi-isometry.
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Geometric mapping theory of the Heisenberg group, sub-Riemannian manifolds, and hyperbolic spaces
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