Pseudo-real Riemann surfaces and chiral regular maps

A Riemann surface is called pseudo-real if it admits anticonformal automorphisms but no anticonformal involution. In this paper, we study general properties of the automorphism groups of such surfaces and their uniformizing NEC groups. In particular, we prove that there exist pseudo-real Riemann surfaces of every possible genus $g\geq2$. We also study pseudo-real surfaces of genus $2$ and $3$. Further, we establish a connection between pseudo-real surfaces with maximal automorphism group, and chiral $3$-valent regular maps, and use this to show there exist such surfaces for infinitely many genera, by exhibiting infinite families of chiral regular maps of type $\{3,k\}$ for all $k \ge 7$.