Given a pair of filling curves α, β on a surface of genus g with n punctures, we explicitly compute the mapping classes realizing the minimal dilatation over all the pseudo-Anosov maps given by the Thurston construction on α,β. We do so by solving for the minimal spectral radius in a congruence subgroup of SL₂(Z). We apply this result to realized lower bounds on intersection number between α and β to give the minimal dilatation over any Thurston construction pA map on Σ_g,n given by a filling pair α ∪ β.

The authors would like to thank Benson Farb for posing the question that motivated this paper, continually supporting them for the duration of the project, and providing extensive comments on this paper. They thank Faye Jackson for her invaluable explanations and intuition, and Amie Wilkinson for teaching a wonderful course in analysis where the authors first began collaborating. The authors would also like to thank Aaron Calderon for his patience in teaching them about entropy, Peter Huxford for his help on Proposition 2.2, Tarik Aougab for helpful remarks on Section 3, Noah Caplinger for discussing Theorem 2.1, and Dan Margalit for general feedback; this paper would not have been possible without their insight.