Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces

We prove that for a quasiself-similar and arcwise connected compact metric space all three known versions of the conformal dimension coincide: the conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular conformal dimension. This answers a question posed by Murugan. Quasisimilar spaces include all approximately self-similar spaces. As an example, the standard Sierpiński carpet is quasiself-similar and thus the three notions of conformal dimension coincide for it. We also give the equality of the three dimensions for combinatorially p-Loewner (CLP) spaces. Both proofs involve using a new notion of combinatorial modulus, which lies between two notions of modulus that have appeared in the literature. The first of these is the modulus studied by Pansu and Tyson, which uses a Carathéodory construction. The second is the one used by Keith and Laakso (and later modified and used by Bourdon, Kleiner, Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches, we gain the flexibility of giving upper bounds for the new modulus from the Pansu–Tyson approach, and the ability of getting lower bounds using the Keith–Laakso approach. Additionally the new modulus can be iterated in self-similar spaces, which is a crucial, and novel, step in our argument.