** Understanding tubular groups **

A *tubular* group is a group that splits as a finite graph of groups with vertex groups and edge groups. I.e. gthe fundamental group of a union of tori and tubes glued along loops.

**Examples**. Right angled Artin groups.

This class contains a number of strange animals.

**Question**. Are free-by-cyclic groups ?

Answer is negative (Gersten 1994). Counterexample is

It is tubular. It is not . Indeed, if it were, the would stabilize a plane, and the actions of , and on that plane would be incompatible.

Here is a non-Hopfian example (Wise):

The morphism , , , is onto but not injective. Nevertheless, this group is .

The isoperimetric behaviour in this class is interesting (Bridson). Chris Cashen can decide when two such groups are quasi-isometric.

Today, I focus on cubulability.

**1. Result **

The geometric intersection number of two geodesics in a flat torus is the total number of intersections, .

An *immersed wall* in a connected 2-dimensional complex is a graph that maps -injectively to and which lifts to a 2-sided embedding in .

An *equitable set* in a tubular space is a collection of curves, one in each vertex torus, that satisfy the equations

**Theorem 1 (Wise)** * A tubular group acts freely on a cube complex iff it has an equitable set. *

This applies to Gersten’s group, to Wise’s non-Hopfian example, to Button’s free-by-cyclic tubular groups.

**Theorem 2 (Woodhouse)** * 1. Given a finite collection of immersed walls , it is decidable wether the corresponding dual cube complex is finite dimensional.*

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2. If the dual is infinite dimensional, then there is an infinite cube (i.e. an infinite chain of nested cubes). *

An instance of infinite dimensional cube is provided by a construction of Rubinstein and Wang.

**Theorem 3 (Woodhouse)** * A tubular group acts freely on a finite dimensional cube complex iff is virtually special. *

Such groups embed is .

The proof consists in a thorough understanding of what the cube complex looks like, and an equivariant map to tree.

**2. Further remarks **

It is likely that examples of automatic, non bi-automatic groups are found in this class. Also, examples with dead ends.

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
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