Actualizing subgroups of 3-manifold groups in homologically small submanifolds

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.