Asymptotic additivity of the Turaev-Viro invariants for a family of $3$-manifolds

TL;DR

This paper demonstrates that the asymptotic behavior of Turaev-Viro invariants is additive under specific gluings of hyperbolic 3-manifold pieces, supporting the volume conjecture for a broad class of manifolds.

Contribution

It proves the additivity of Turaev-Viro invariants under certain gluings, extending the volume conjecture to new families of 3-manifolds.

Findings

Asymptotics of Turaev-Viro invariants are additive under specific gluings.

Constructed families of manifolds satisfying the extended volume conjecture.

Extended the applicability of the Turaev-Viro invariant volume conjecture.

Abstract

In this paper, we show that the Turaev-Viro invariant volume conjecture posed by Chen and Yang is preserved under gluings of toroidal boundary components for a family of $3$-manifolds. In particular, we show that the asymptotics of the Turaev-Viro invariants are additive under certain gluings of elementary pieces arising from a construction of hyperbolic cusped $3$-manifolds due to Agol. The gluings of the elementary pieces are known to be additive with respect to the simplicial volume. This allows us to construct families of manifolds with an arbitrary number of hyperbolic pieces such that the resultant manifolds satisfy an extended version of the Turaev-Viro invariant volume conjecture.