# Guts and volume for hyperbolic $3$-orbifolds with underlying space $S^3$

## TL;DR

This paper establishes a lower volume bound for hyperbolic 3-orbifolds with underlying space S^3 containing essential 2-suborbifolds, using guts analysis and topological techniques.

## Contribution

It introduces a method to compute the guts of such orbifolds and characterizes those with empty guts, advancing understanding of their geometric structure.

## Key findings

- Lower volume bound for specific hyperbolic 3-orbifolds.
- Characterization of orbifolds with empty guts.
- Topological analysis techniques for orbifold guts.

## Abstract

For a hyperbolic $3$-orbifold with underlying space the $3$-sphere, we obtain a lower bound on its volume in the case that it contains an essential $2$-suborbifold with underlying space the $2$-sphere with four cone points. Our techniques involve computing the guts of the orbifold split along the $2$-suborbifold via a careful analysis of its topology. We also characterize the orbifolds of this type that have empty guts.