Hyperbolization procedures

A hyperbolization procedure is a procedure that turns a polyhedral complex ${\displaystyle K}$ into a non-positively curved space ${\displaystyle {\mathcal {H}}(K)}$, retaining some of its topological features. Roughly speaking, the procedure consists in replacing every cell of ${\displaystyle K}$ with a copy of a certain non-positively curved manifold with boundary, which is fixed a priori and is called the hyperbolizing cell of the procedure.

There are many different hyperbolization procedures available in the literature. While they all satisfy some common axioms, they differ by what kind of polyhedral complex is allowed as input and what kind of hyperbolizing cell is used. As a result, different procedures preserve different topological features and provide spaces with different geometric flavors. The first hyperbolization procedures were introduced by Mikhael Gromov in ^[1] and later other versions were developed by several mathematicians including Ruth Charney, Michael W. Davis, and Pedro Ontaneda.

It is important to note that the word "hyperbolization" here does not have the same meaning that it has in the uniformization or hyperbolization results typical of low-dimensional geometry. Indeed, the space ${\displaystyle {\mathcal {H}}(K)}$ is not homeomorphic to ${\displaystyle K}$. For instance, ${\displaystyle {\mathcal {H}}(K)}$ is always aspherical, regardless of whether ${\displaystyle K}$ is aspherical. Moreover, despite the name of the procedure, ${\displaystyle {\mathcal {H}}(K)}$ is not always guaranteed to be negatively curved, so some authors refer to these procedures as asphericalization procedures.