A median space is a metric space $X$ for which for any three points $x, y , z \in X $ there exists a unique point $m$ such that $d(x,m)+ d(m, y)= d(x , y ), d(x,m)+ d(m, z)= d(x , z ), d(y,m)+ d(m, z)= d(y , z )$. $CAT(0)$-polygonal complexes are simply connected collections of glued (on their faces) polyhedra of varying dimension such that each link is flag. Does there exist a metric on $CAT(0)$-polygonal complexes such that they are median spaces (in an analogous fashion to $CAT(0)$-cube complexes)?
The question is vague. The median metric defined on CAT(0) cube complexes satisfy two fundamental properties:
- For finite-dimensional cube complexes, it is bi-Lipschitz equivalent to the CAT(0) metric.
- Any automorphism of a cube complex preserves both the CAT(0) and the median metrics.
What I claim is that there exist finite-dimensional CAT(0) polyhedral complexes which do not admit a median metric satisfying the above two points. Indeed, otherwise any group acting geometrically on such a polyhedral complex would act geometrically on a median space. By combining the following observation with the fact that any group acting on a median space with an unbounded orbit does not satisfy Kazhdan's property (T), we conclude that it is far from being true.
Theorem: There exists an infinite group satisfying Kazhdan's property (T) which acts geometrically on a finite-dimensional CAT(0) polyhedral complex.
My guess is that there exist plenty of such complexes. For one example, you can see Caprace's paper A sixteen-relator presentation of an infinite hyperbolic Kazhdan group. There, the polyhedral complex is even CAT(-1). [See Yves' comment below for additional information.]
I suspect that the second point above can be removed, ie., there should exist finite-dimensional CAT(0) polyhedral complexes which do not admit a bi-Lipschitz equivalent median metric.