On Cannon cone types and vector-valued multiplicative functions for genus-two surface group

We consider Cannon cone types for a surface group of genus g, and we give algebraic criteria for establishing the cone type of a given cone and of all its sub-cones. We also re-prove that the number of cone types is exactly 8g(2g - 1)+1. In the genus 2 case, we explicitly provide the 48x 48 matrix of cone types, and we prove that it is primitive, hence Perron-Frobenius. Finally we define vector-valued multiplicative functions and we show how to compute their values by means of the matrix of cone types.