Generalized progressions in the integer Heisenberg group have VC-dimension at most 267, and in the free group on k generators at most 3k-1.
On the Vapnik-Chervonenkis dimension of products of intervals in $\mathbb{R}^d$
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abstract
We study combinatorial complexity of certain classes of products of intervals in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the Vapnik-Chervonenkis dimension of the set of balls in $\ell_\infty^d$ -- which denotes $\R^d$ equipped with the sup norm -- equals $\lfloor (3d+1)/2\rfloor$.
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VC-dimension of generalized progressions in some nonabelian groups
Generalized progressions in the integer Heisenberg group have VC-dimension at most 267, and in the free group on k generators at most 3k-1.