Regulators, entropy and infinite determinants

TL;DR

This paper explores the connections between regulator maps, entropy, and determinants in group von Neumann algebras, highlighting both real and p-adic cases, and surveys known results with some new insights.

Contribution

It provides a comprehensive survey of the relations between regulators, entropy, and determinants, including new results in section 3.1 and discussions on p-adic analogues.

Findings

Regulator values correspond to topological entropy of group actions.

Determinants in von Neumann algebras relate to entropy in both real and p-adic contexts.

The paper identifies open questions and future research directions.

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.