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\begin{document}$${\mathbb{G}}$$\end{document} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{F}_{m,r}}$$\end{document} denote the free Lie group of step r and m generators. Let also \documentclass[12pt]{minimal}
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\begin{document}$${\pi:\mathbb{F}_{m,r}\to\mathbb{G}}$$\end{document} be a lifting map. We show that any horizontally convex function u on \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{G}}$$\end{document} lifts to a horizontally convex function \documentclass[12pt]{minimal}
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\begin{document}$${u\circ \pi}$$\end{document} on \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{F}_{m,r}}$$\end{document} (with respect to a suitable horizontal frame on \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{F}_{m,r}}$$\end{document}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}=\sum_{j=1}^m X_j^2}$$\end{document} on a Carnot group of step two such that the relevant \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document}-gauge function d (i.e., d2-Q is the fundamental solution for \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document}) is not h-convex with respect to the horizontal frame {X1, . . . , Xm}. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).
