The existence of a quaternion algebra construction of Hurwitz surfaces is due to G. Shimura ('67). An explicit presentation was described by N. Elkies in a pair of articles ('98 and '99). R. Vogeler applied group-theoretic methods to study Hurwitz surfaces in '03, and obtained differential-geometric, and systolic, consequences. In particular, he calculated the number of systolic loops in each genus below a million. In '07, Kobi Gurkan developed magma models for the Hurwitz quaternion order and for Hurwitz surfaces of genus 3 and 14.

Hurwitz surfaces of genus 14: see wikipedia page "First Hurwitz triplet"

Hurwitz surface of genus 3: see wikipedia page "Klein quartic"