Trace Systoles and Sink Constants

Let [Formula: see text] be a surface of finite type with [Formula: see text], and a representation [Formula: see text] from the fundamental group [Formula: see text] into [Formula: see text]. We define the trace systole of [Formula: see text], denoted [Formula: see text] as follows: [Formula: see text] When [Formula: see text] is endowed with a hyperbolic structure, the trace systole of the holonomy representation is naturally related to the usual systolic length of the hyperbolic surface, which is one of the motivations for this study. The function [Formula: see text] is bounded above on relative character varieties of [Formula: see text], and in this article we compute explicitly the optimal bounds for the one-holed torus, the four-holed sphere and the non-orientable surface of genus [Formula: see text]. The proofs rely on the correspondance between representations of these surface groups and so-called Markoff maps which were introduced by Bowditch. From this, we infer various consequences on the optimal systolic inequalities of certain hyperbolic manifolds and also on non-Fuchsian representations for these surfaces.