Actualizing subgroups of 3-manifold groups in homologically small submanifolds

TL;DR

This paper proves that certain subgroups of 3-manifold groups can be realized within specific submanifolds with controlled topology, providing bounds on their genus and homology, under particular boundary and homological conditions.

Contribution

It establishes new bounds on subgroups of 3-manifold groups, linking algebraic properties to geometric realizations with explicit topological constraints.

Findings

Subgroups are carried by submanifolds with incompressible boundary.

Euler characteristic of submanifolds is bounded below by 1 minus the dimension of homology.

Provides bounds on the genus of boundary components based on homological data.

Abstract

Let $Y$ be a simple $3$-manifold, and let $A$ be a finitely generated, freely indecomposable subgroup of $\pi_1(Y)$. Set $\eta=\dim H_1(A;{\bf F}_2)$. Suppose that either (a) $\partial Y\ne\emptyset$ or (b) $\dim H_1(Y;{\bf F}_2)\ge3\eta^2-4\eta+4$. Under these hypotheses, we show that $A$ is carried by some compact, connected three-dimensional submanifold $Z$ of $\text{int} \;Y$ such that (1) $\partial Z$ is non-empty, and each of its components is incompressible in $Y$; (2) the Euler characteristic of $Z$ is bounded below by $1-\eta$; and (3) $\dim H_1(Z;{\bf F}_2)\le 3\eta^2-4\eta+1$. The conclusion implies that any boundary component of $Z$ is an incompressible surface of genus at most $\eta$. In Case (b), this should be compared with earlier results proved by Agol-Culler-Shalen and Culler-Shalen, which provide a surface of genus at most $\eta$ under weaker hypotheses (the lower bound on $\dim H_1(Y; {\bf F}_2)$ being linear in $\eta$ rather than quadratic), but do not give any relationship between the given subgroup $A$ and this surface. In a forthcoming paper we will apply the result to give a new upper bound for the ratio of the rank of the mod 2 homology of a closed, orientable hyperbolic $3$-manifold to the volume of the manifold.