  We formulate the geometric P=W conjecture for singular character varieties.
We establish it for compact Riemann surfaces of genus one, and obtain partial
results in arbitrary genus. To this end, we employ non-Archimedean, birational
and degeneration techniques to study the topology of the dual boundary complex
of certain character varieties. We also clarify the relation between the
geometric and the cohomological P=W conjectures.
