Spectral Dynamics and Fuglede--Kadison Determinants for Wreath Products
Abstract
We investigate connections between operator spectra, complex dynamics, and determinants for wreath product groups. For a nontrivial finite abelian group G, the natural action of Gwrℤ on a rooted tree produces a family of operators with a recursive structure. Applying Schur complements to this family yields a rational map on complex projective space. We determine the projective spectrum--the parameter values for which the operator pencil fails to be invertible--as the union of the Julia set of this map and an explicitly described subset of its Fatou set. We also prove that the associated Koopman representation and the left regular representation are weakly equivalent and therefore give the same projective spectrum. We then compute Fuglede--Kadison determinants for a family of operators on more general wreath products. These formulas yield injective operators with zero determinant for wreath products Kwrℤ, where K is a discrete group admitting a nontrivial projection e∈ℂ[K]. This answers negatively a question of Li and Thom about whether injectivity implies positive determinant for operators defined by matrices over complex group rings of amenable groups. For Gtfrtimesσℤ, where Gtf is any countable torsion-free abelian group and σ∈Aut(Gtf), we prove that injectivity does imply positive determinant in every matrix size.
Related
- Fuglede--Kadison Determinants of Shifted Matrix Semicircles
- Metric Spectral Geometry of ℤd-Crossed Products beyond Equicontinuity
- Large deviations of spectral determinants of matrix-valued random Schrödinger operators and Dyson Brownian motion in cubic potentials
- Fuglede's theorem in the Heisenberg group
- A note on the non-wandering dynamics in the n-fold symmetric products
BibTeX
@misc{wang2026spectral,
title = {{Spectral Dynamics and Fuglede--Kadison Determinants for Wreath Products}},
author = {Penghui Wang and Xumin Wang and Rongwei Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03711},
eprint = {2611.03711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03711},
}
RIS
TY - UNPB
TI - Spectral Dynamics and Fuglede--Kadison Determinants for Wreath Products
AU - Penghui Wang
AU - Xumin Wang
AU - Rongwei Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03711
UR - https://arxiv.org/abs/2611.03711
AB - We investigate connections between operator spectra, complex dynamics, and determinants for wreath product groups. For a nontrivial finite abelian group G, the natural action of Gwrℤ on a rooted tree produces a family of operators with a recursive structure. Applying Schur complements to this family yields a rational map on complex projective space. We determine the projective spectrum--the parameter values for which the operator pencil fails to be invertible--as the union of the Julia set of this map and an explicitly described subset of its Fatou set. We also prove that the associated Koopman representation and the left regular representation are weakly equivalent and therefore give the same projective spectrum. We then compute Fuglede--Kadison determinants for a family of operators on more general wreath products. These formulas yield injective operators with zero determinant for wreath products Kwrℤ, where K is a discrete group admitting a nontrivial projection e∈ℂ[K]. This answers negatively a question of Li and Thom about whether injectivity implies positive determinant for operators defined by matrices over complex group rings of amenable groups. For G_tfrtimes_σℤ, where G_tf is any countable torsion-free abelian group and σ∈Aut(G_tf), we prove that injectivity does imply positive determinant in every matrix size.
ER -