Regulators, entropy and infinite determinants

Abstract

In this note we describe instances where values of the $K$-theoretical regulator map evaluated on topological cycles equal entropies of topological actions by a group $\Gamma$. These entropies can also be described by determinants on the von Neumann algebra of $\Gamma$. The relations were first observed for real regulators. The latter have $p$-adic analogues and both $p$-adic entropy and $p$-adic determinants were then defined so that similar relations hold as in the real case. We describe this $p$-adic theory in the second part of the paper. This note is almost entirely a survey of known results with the exception of some results in section 3.1. However the different aspects of the theory have not been discussed together before. Along the way we point out several open questions and possible directions for further research.