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

TL;DR

This paper analyzes Cannon cone types for genus-g surface groups, providing algebraic criteria, explicit matrices for genus two, and methods to compute vector-valued multiplicative functions using Perron-Frobenius theory.

Contribution

It introduces algebraic criteria for cone types, explicitly constructs the cone type matrix for genus two, and connects these to vector-valued multiplicative functions.

Findings

Number of cone types is exactly 8g(2g - 1)+1.

The cone type matrix for genus two is a primitive 48x48 matrix.

Methods to compute vector-valued multiplicative functions using the matrix.

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 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 $48\times 48$ matrix of cone types, $M,$ and we prove that $M$ is primitive, hence Perron-Frobenius. Finally we define vector-valued multiplicative functions and we show how to compute their values by means of $M$.