A note on Kakeya sets of horizontal and SL(2)$SL(2)$ lines

We consider unions of S L ( 2 ) $SL(2)$ lines in R 3 $\mathbb {R}^{3}$ . These are the form = a , b 0 + span c d 1 $$\begin{equation*} \hspace*{5pc} (a,b,0) \mathrm{span}(c,d,1), \end{equation*}$$ where − $ad - bc 1$ show that if $\mathcal {L}$ is Kakeya set lines, then union ∪ $\cup \mathcal has Hausdorff dimension 3. This answers question Wang and Zahl. The can be identified with horizontal first Heisenberg group, we obtain main result as corollary more general statement concerning lines. established via point-line duality principle between conical combined recent work on restricted families projections to planes, due Gan, Guo, Guth, Harris, Maldague Wang. Our also for Nikodym sets associated which special case Kim.