Jones' representations of R. Thompson's groups not induced by finite-dimensional ones

TL;DR

This paper constructs specific Pythagorean unitary representations of Richard Thompson's group F from Hilbert space isometries, demonstrating that some cannot be derived from finite-dimensional induced representations, a novel finding in the field.

Contribution

It introduces a condition on isometries that guarantees the associated representation is not induced by any finite-dimensional representation, marking the first such result.

Findings

Most representations parametrized by the 3-sphere have this property.

Two sub-circles of the parameter space do not satisfy this property.

Provides explicit construction and criteria for non-finite-dimensional induced representations.

Abstract

Given any linear isometry from a Hilbert space to its square one can explicitly construct a so-called Pythagorean unitary representation of Richard Thompson's group F. We introduce a condition on the isometry implying that the associated representation does not contain any induced representations by finite-dimensional ones. This provides the first result of this kind. We illustrate this theorem via a family of representations parametrised by the real 3-sphere for which all of them have this property except two sub-circles.