In this paper, we construct Holder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $mathbb{H}$. Pansu and Gromov observed that any surface embedded in $mathbb{H}$ has Hausdorff dimension at least 3, so there is no $alpha$-Holder embedding of a surface into $mathbb{H}$ when $alpha>frac{2}{3}$. Zust improved this result to show that when $alpha>frac{2}{3}$, any $alpha$-Holder map from a simply-connected Riemannian manifold to $mathbb{H}$ factors through a metric tree. In the present paper, we show that Zusts result is sharp by constructing $(frac{2}{3}-epsilon)$-Holder maps from $D^2$ and $D^3$ to $mathbb{H}$ that do not factor through a tree. We use these to show that if $0<alpha < frac{2}{3}$, then the set of $alpha$-Holder maps from a compact metric space to $mathbb{H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $mathbb{R}^3$ to $mathbb{H}$ with Holder exponents arbitrarily close to $frac{2}{3}$.
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