A relation between the Baseilhac-Benedetti and the Bonahon-Liu-Wong-Yang invariants

TL;DR

This paper explores the deep connections between various invariants of pseudo-Anosov homeomorphisms and hyperbolic 3-manifolds at roots of unity, aiming to establish precise relationships among them.

Contribution

It provides a rigorous proof of the relations between the Baseilhac-Benedetti invariants, Bonahon-Liu-Wong-Yang invariants, and abelian $rak{gl}_1$-invariants at roots of unity.

Findings

Established a precise relation between Baseilhac-Benedetti and Bonahon-Liu-Wong-Yang invariants.

Connected these invariants to abelian $rak{gl}_1$-invariants.

Enhanced understanding of the structure of invariants at roots of unity.

Abstract

Baseilhac-Benedetti, following ideas of Kashaev, introduced invariants of pseudo-Anosov homeomorphisms of punctured hyperbolic surfaces that depend on a complex root of unity of odd order. Around the same time, Bonahon-Liu introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. A little later, Dimofte and the first author introduced invariants of cusped hyperbolic 3-manifolds at roots of unity using their geometric representation. In another effort, Bonahon-Wong-Yang introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. All these invariants are conjecturally closely related, and our aim is to prove a precise relation between the Baseilhac-Benedetti invariants, the Bonahon-Liu-Wong-Yang and the lesser-known abelian $\mathfrak{gl}_1$-invariants.