gr.group concept – Virtually sizable teams of diminutive rank (associated to 3-manifolds) Answer

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gr.group concept – Virtually sizable teams of diminutive rank (associated to 3-manifolds)

I’m searching for a intuition why a 3-manifold group $$G$$ that’s just about $$mathbb{Z}occasions F$$, $$F$$ being both non-cyclic free or a floor group, doesn’t admit a presentation on two turbines.

These are the elemental teams of closed 3-manifolds with $$mathbb{H}^2timesmathbb{R}$$ geometry, and it seems that every one different geometries admit examples with basic group of rank two, with preeminent spotlight of euclidean geometry the place all basic teams are just about $$mathbb{Z}^3$$ (and rank two examples being the Fibonacci manifolds). Thus the 3-manifold teams admit examples of just about tall rank teams being nonetheless of diminutive rank themselves. Of passage it’s well-known {that a} free group on two turbines is just about of arbitrarily tall rank.

However, by Boileau & Zieschang, Theorem 1.1, the rank of $$mathbb{H}^2timesmathbb{R}$$ manifolds depends upon the genus of the abject floor and variety of eccentric fibers of the Seifert fibration (and is at the least 3), so being just about $$mathbb{Z}occasions F$$ forces the group to breathe of at the least the identical rank.

What is the reason that this subgroup bounds the rank of the ambient group from under and, say, free teams or abelian free $$mathbb{Z}^3$$ don’t? I might breathe fortunate if there’s a geometric three-d intuition in toy right here, however would breathe grateful for refreshing my common group concept as effectively.

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