# On spectral measures for certain unitary representations of R. Thompson's group F

## TL;DR

This paper studies spectral measures associated with unitary representations of Thompson's group F derived from an anyonic quantum spin chain, revealing their absolute continuity and providing computational methods.

## Contribution

It introduces a method to compute spectral measures for Thompson's group F representations and characterizes their absolute continuity, extending to Brown-Thompson groups.

## Key findings

- Spectral measures are mostly absolutely continuous with respect to Lebesgue measure.
- A method to calculate spectral measures for elements of F is provided.
- Results extend to Brown-Thompson groups F_n.

## Abstract

The Hilbert space $\mathcal H$ of backward renormalisation of an anyonic quantum spin chain affords a unitary representation of Thompson's group $F$ via local scale transformations. Given a vector in the canonical dense subspace of $\mathcal H$ we show how to calculate the corresponding spectral measure for any element of $F$ and illustrate with some examples. Introducing the "essential part" of an element we show that the spectral measure of any vector in $\mathcal H$ is, apart from possibly finitely many eigenvalues, absolutely continuous with respect to Lebesgue measure. The same considerations and results hold for the Brown-Thompson groups $F_n$ (for which $F=F_2$).