Three-dimensional simplicial gravity and combinatorics of group presentations

TL;DR

This paper connects simplicial quantum gravity with combinatorial group theory, showing exponential growth of 3-manifolds and proposing a 3D gravity model with phase structure insights.

Contribution

It introduces a novel approach linking simplicial gravity problems to combinatorial group theory and proposes a new 3D gravity model with phase structure analysis.

Findings

Number of 3-manifolds grows exponentially with tetrahedra

Proposed a 3D gravity model with scalar fermions

Qualitative phase structure compatible with experiments

Abstract

We demonstrate how some problems arising in simplicial quantum gravity can be successfully addressed within the framework of combinatorial group theory. In particular, we argue that the number of simplicial 3-manifolds having a fixed homology type grows exponentially with the number of tetrahedra they are made of. We propose a model of 3D gravity interacting with scalar fermions, some restriction of which gives the 2-dimensional self-avoiding-loop-gas matrix model. We propose a qualitative picture of the phase structure of 3D simplicial gravity compatible with the numerical experiments and available analytical results.