Actualizing subgroups of 3-manifold groups in homologically small submanifolds
Rosemary K. Guzman, Peter B. Shalen
TL;DR
The paper addresses how to realize finitely generated, freely indecomposable subgroups $A$ of a simple 3-manifold group $\pi_1(Y)$ as carried by a homologically small submanifold. It develops a global, tower-based framework utilizing characteristic submanifolds, books of $I$-bundles, and acylindrical splittings to push $A$ into a compact submanifold $Z$ inside $\mathrm{int}Y$, with boundary incompressible in $Y$ and explicit quadratic bounds on homology: $0\le \overline\chi(Z)\le \eta-1$ and $\dim H_1(Z;\bf F_2)\le 3\eta^2-4\eta+1$, where $\eta=\dim H_1(A;\bf F_2)$. The approach hinges on constructing good towers of two-sheeted coverings, lifting maps to the tower, and then pushing down to submanifolds $T$ in a base manifold, maintaining control of Euler characteristics and rank bounds at each step. The main technical engine is a push-down lemma in a two-sheeted cover, which provides the recursion step that yields the desired bound and the connection between $A$ and an incompressible boundary surface of genus at most $\eta$. The results generalize earlier linear-bounds results and set the stage for a forthcoming bound relating the mod 2 homology rank to hyperbolic volume, highlighting a deep link between subgroup structure and geometric/topological size in 3-manifolds.
Abstract
Let $Y$ be a simple $3$-manifold, and let $A$ be a finitely generated, freely indecomposable subgroup of $π_1(Y)$. Set $η=\dim H_1(A;{\bf F}_2)$. Suppose that either (a) $\partial Y\ne\emptyset$ or (b) $\dim H_1(Y;{\bf F}_2)\ge3η^2-4η+4$. 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 $1-η$; and (3) $\dim H_1(Z;{\bf F}_2)\le 3η^2-4η+1$. The conclusion implies that any boundary component of $Z$ is an incompressible surface of genus at most $η$. 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 $η$ under weaker hypotheses (the lower bound on $\dim H_1(Y; {\bf F}_2)$ being linear in $η$ 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.