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

### Volume 158 / 2019

Colloquium Mathematicum 158 (2019), 77-89
MSC: Primary 20F67; Secondary 05C25, 43A65.
DOI: 10.4064/cm7589-9-2018
Published online: 24 June 2019

#### Abstract

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 subcones. 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 $48\times 48$ 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.