[Paper Review] Adding high powered relations to large groups

Marc Lackenby|ArXiv.org|Dec 15, 2005

Geometric and Algebraic Topology2 references4 citations

TL;DR

This paper establishes that for any finitely generated large group $G$ and any finite set of elements $g_1, \dots, g_r \in G$, the quotient $G / \langle\!\langle g_1^n, \dots, g_r^n \rangle\!\rangle$ remains large for infinitely many integers $n$, particularly when $n$ is a sufficiently large multiple of the index $[G:H]$ for a finite-index normal subgroup $H$ mapping onto a non-abelian free group. The result extends to free groups, where largeness holds for all but finitely many $n$, and has strong applications to 3-manifold topology, particularly in Dehn surgery and the persistence of largeness across infinitely many filling slopes.

ABSTRACT

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. The main theorem of the paper is as follows. Let G be a finitely generated, large group and let g_1,...,g_r be a collection of elements of G. Then G/<> is also large, for infinitely many integers n. Furthermore, when G is free, this holds for all but finitely many n. These results have the following application to Dehn surgery. Let M be a compact orientable 3-manifold with boundary a torus. Suppose that the 3-manifold obtained by Dehn filling some slope on the boundary has large fundamental group. Then this is true for infinitely many filling slopes.

Motivation & Objective

• To establish conditions under which quotienting a large group by high powers of its elements preserves the property of largeness.
• To extend known results on hyperbolicity and largeness under quotienting by high powers, particularly in the context of group theory and 3-manifold topology.
• To provide a topological proof of largeness using covering space theory and cohomological criteria, especially via Theorem 1.3 on cohomology maps.
• To apply the main result to Dehn surgery, showing that if one Dehn filling yields a manifold with large fundamental group, then infinitely many fillings do so as well.

Proposed method

• Uses the topological realization of free groups as fundamental groups of bouquets of circles, and constructs 2-complexes by attaching 2-cells along loops representing $g_i^n$.
• Applies Theorem 1.3, which gives a cohomological criterion for largeness: if the inclusion-induced maps on $H^1(-; \mathbb{F}_p)$ fail to be injective (with a dimension condition for $p=2$), then the fundamental group is large.
• Reduces Theorem 1.1 to Theorem 1.2 by embedding a large group $G$ into a free group via finite-index subgroups and analyzing the image of elements under power maps.
• Employs the structure of Dehn surgery on links in 3-manifolds, expressing the fundamental group of the resulting manifold as a quotient of the original group modulo relations involving meridians and filling slopes.
• Uses the distance $\Delta(\mu_i, s_i)$ between meridians and filling slopes to control the exponents in the quotient, linking topological distance to algebraic largeness.
• Applies Theorem 1.1 to the quotient group $\pi_1(M)/\langle\!\langle \lambda_1^N, \dots, \lambda_r^N \rangle\!\rangle$, showing it is large for infinitely many $N$, hence the resulting fundamental group is large.

Experimental results

Research questions

• RQ1 Under what conditions does quotienting a large group by high powers of its elements preserve the property of largeness?
• RQ2 Can the result that $F/\langle\!\langle g_1^n, \dots, g_r^n \rangle\!\rangle$ is large for all but finitely many $n$ in a free group be extended to general large groups?
• RQ3 Does the largeness of the fundamental group persist across infinitely many Dehn fillings of a 3-manifold, given that it holds for one filling?
• RQ4 What topological or algebraic conditions ensure that the fundamental group of a 3-manifold obtained by Dehn surgery remains large?

Key findings

• For any finitely generated large group $G$ and elements $g_1, \dots, g_r$, the quotient $G / \langle\!\langle g_1^n, \dots, g_r^n \rangle\!\rangle$ is large for infinitely many integers $n$, particularly when $n$ is a sufficiently large multiple of $[G:H]$ for a finite-index normal subgroup $H$ mapping onto a non-abelian free group.
• In the case of a finitely generated non-abelian free group $F$, the quotient $F / \langle\!\langle g_1^n, \dots, g_r^n \rangle\!\rangle$ is large for all but finitely many $n$, providing a stronger result than in the general case.
• Theorem 1.1 implies that for a knot $K$ in $S^3$, the $n$-fold cyclic branched cover has large fundamental group for infinitely many $n$, even though it may not hold for all $n$.
• Theorem 3.1 establishes that if $M$ is a compact orientable 3-manifold with large fundamental group and $L$ is a link in $M$, then Dehn surgery along slopes $s_i$ with $\Delta(\mu_i, s_i)$ divisible by a fixed $N$ yields a manifold with large fundamental group for infinitely many such slope collections.
• Theorem 1.5 follows: if one Dehn filling of a 3-manifold with toroidal boundary yields a large fundamental group, then this holds for infinitely many distinct filling slope collections.