Extremal entropy for products of Fuchsian representations

Abstract: In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of $\mathsf{PO}(2, 1) \times \mathsf{PO}(2, 1)$, its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination $λ$ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in $λ$. We hope this will be a starting point for a more general study of extremal entropy.