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There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm guessing they will soon be proven in full generality). The proof is lucidly discussed in Danny Calegari's blog. The theorems state that every (compact, compact orientable , irreducible ) 3-manifold with infinite fundamental group has a finite (not necessarily regular) cover which is Haken or a surface bundle over a circle, correspondingly. This implies various good things for a 3-manifold with fundamental group π, including:

1. π is large, meaning that π has a finite index subgroup which maps onto a free group with at least 2 generators. In particular the Betti numbers of finite covers can become arbitrarily large.
2. π is linear over $\mathbb{Z}$, i.e. π admits a faithful representation $\pi \to \mathrm{GL}(n,\mathbb{Z})$ for some $n$. (Thurston conjectured that $n\leq 4$ is sufficient).
3. π is virtually biorderable.

Stefan Friedl, from whose comment the above list is an excerpt, summarizes the situation as follows:

It seems like every nice property of fundamental groups which one can possibly ask for either holds for π or a finite index subgroup of π.

All well and good. But how could you sell' that to somebody who isn't a classically-oriented 3-dimensional topologist? An elevator pitch is defined by Wikipedia as follows:

An elevator pitch is a short summary used to quickly and simply define a product, service, or organization and its value proposition. The name "elevator pitch" reflects the idea that it should be possible to deliver the summary in the time span of an elevator ride, or approximately thirty seconds to two minutes. In The Perfect Elevator Speech, Aileen Pincus states that an elevator speech should "sum up unique aspects of your service or product in a way that excites others."

The Virtual Fibering Conjecture (or the Virtual Haken Conjecture) was the grand conjecture in 3-manifold topology following Geometrization, and thus must have/ should have/ ought to have (I believe) a compelling elevator pitch. For contrast, Geometrization is easy to sell' because it directly applies to the Homeomorphism Problem in 3-manifold topology: Given two 3-manifolds, determine whether or not they are homeomorphic. Geometrization allows you to canonically decompose both manifolds into submanifolds with geometric structure, and then to compare geometric invariants. In terms of "The Goals of Mathematical Research" as given in the introduction to The Princeton Companion to Mathematics, this corresponds to the goal of Classifying.

Question: What is a good elevator pitch for Virtual Fibering (or for Virtual Haken), explaining the utility of these results in terms of "the fundamental goals of mathematical research" (Solving Equations, Classifying, Generalizing, Discovering Patterns, Explaining Patterns and Coincidences, Counting and Measuring, and Finding Explicit Algorithms). The target would be mathematicians who are not 3-dimensional topologists.

Everyone in the approximate vicinity of the field instinctively feels that these are historic results, but I'd like to be able to justify that feeling (in the abovementioned sense) to myself and to others.

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# Elevator pitch for the Virtual Fibering Theorem

There has been a great deal of excitement among topologists about the proof of the Virtual Haken Theorem, and in fact of the Virtual Fibering Theorem (for closed hyperbolic 3-manifolds, but I'm guessing they will soon be proven in full generality). The proof is lucidly discussed in Danny Calegari's blog. The theorems state that every (compact, orientable, irreducible) 3-manifold has a finite (not necessarily regular) cover which is Haken or a surface bundle over a circle, correspondingly. This implies various good things for a 3-manifold with fundamental group π, including:

1. π is large, meaning that π has a finite index subgroup which maps onto a free group with at least 2 generators. In particular the Betti numbers of finite covers can become arbitrarily large.
2. π is linear over $\mathbb{Z}$, i.e. π admits a faithful representation $\pi \to \mathrm{GL}(n,\mathbb{Z})$ for some $n$. (Thurston conjectured that $n\leq 4$ is sufficient).
3. π is virtually biorderable.

Stefan Friedl, from whose comment the above list is an excerpt, summarizes the situation as follows:

It seems like every nice property of fundamental groups which one can possibly ask for either holds for π or a finite index subgroup of π.

All well and good. But how could you sell' that to somebody who isn't a classically-oriented 3-dimensional topologist? An elevator pitch is defined by Wikipedia as follows:

An elevator pitch is a short summary used to quickly and simply define a product, service, or organization and its value proposition. The name "elevator pitch" reflects the idea that it should be possible to deliver the summary in the time span of an elevator ride, or approximately thirty seconds to two minutes. In The Perfect Elevator Speech, Aileen Pincus states that an elevator speech should "sum up unique aspects of your service or product in a way that excites others."

The Virtual Fibering Conjecture (or the Virtual Haken Conjecture) was the grand conjecture in 3-manifold topology following Geometrization, and thus must have/ should have/ ought to have (I believe) a compelling elevator pitch. For contrast, Geometrization is easy to sell' because it directly applies to the Homeomorphism Problem in 3-manifold topology: Given two 3-manifolds, determine whether or not they are homeomorphic. Geometrization allows you to canonically decompose both manifolds into submanifolds with geometric structure, and then to compare geometric invariants. In terms of "The Goals of Mathematical Research" as given in the introduction to The Princeton Companion to Mathematics, this corresponds to the goal of Classifying.

Question: What is a good elevator pitch for Virtual Fibering (or for Virtual Haken), explaining the utility of these results in terms of "the fundamental goals of mathematical research" (Solving Equations, Classifying, Generalizing, Discovering Patterns, Explaining Patterns and Coincidences, Counting and Measuring, and Finding Explicit Algorithms). The target would be mathematicians who are not 3-dimensional topologists.

Everyone in the approximate vicinity of the field instinctively feels that these are historic results, but I'd like to be able to justify that feeling (in the abovementioned sense) to myself and to others.