Actualizing subgroups of 3-manifold groups in homologically small submanifolds
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912228524949504 |
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| author | Guzman, Rosemary K. Shalen, Peter B. |
| author_facet | Guzman, Rosemary K. Shalen, Peter B. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Actualizing subgroups of 3-manifold groups in homologically small submanifolds Guzman, Rosemary K. Shalen, Peter B. Geometric Topology 57K32 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. |
| title | Actualizing subgroups of 3-manifold groups in homologically small submanifolds |
| topic | Geometric Topology 57K32 |
| url | https://arxiv.org/abs/2502.07122 |