2025/02/10 by Guzman, Rosemary K., Shalen, Peter B.
#57K32 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2502.07122
Let Y be a simple 3-manifold, and let A be a finitely generated, freely indecomposable subgroup of π1(Y). Set η=dim H1(A;\bf F2). Suppose that either (a) ∂ Y≠∅ or (b) dim H1(Y;\bf F2)≥3η2-4η+4. Under these hypotheses, we show that A is carried by some compact, connected three-dimensional submanifold Z of int Y such that (1) ∂ 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 H1(Z;\bf F2)≤ 3η2-4η+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 H1(Y; \bf F2) 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.