The Group of Symmetric Euler Characteristic -3 Coy L. May and Jay Zimmerman

Let G be a finite group. The symmetric Euler characteristic χ(G) is the maximal Euler characteristic of any surface X (orientable or nonorientable) on which G acts. The groups of symmetric Euler characteristic χ ≥ −2 have been classified. We show that S5 is the unique group with Euler characteristic -3. Two related parameters are the symmetric genus σ and the symmetric crosscap number σ̃. We consider some basic relations among the three parameters and also determine σ̃(Zn×Zn) when n is even. The quantity σ̃ − 2σ is a measure of the difference between the orientable genus action and the non-orientable genus action. We present a series of examples that show this quantity can be either positive or negative and have arbitrarily large magnitude in either case.