Notes of Daniel Woodhouse’s Cambridge lecture 10-01-2017

Understanding tubular groups

A tubular group is a group that splits as a finite graph of groups with {{\mathbb Z}^2} vertex groups and {{\mathbb Z}} 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 {CAT(0)} ?

Answer is negative (Gersten 1994). Counterexample is

\displaystyle  \begin{array}{rcl}  G&=&\langle a,b,c,t|a^t=a,\,b^t=ba,\,c^t=ca^2\rangle\\ &=&\langle x,y,u,v|[x,y]=1,\,(yx)^u=y,\,(yx)^v=yx^{-1}\rangle. \end{array}

It is tubular. It is not {CAT(0)}. Indeed, if it were, the {{\mathbb Z}^2} would stabilize a plane, and the actions of {x},{xy} and {xy^{-1}} on that plane would be incompatible.

Here is a non-Hopfian example (Wise):

\displaystyle  G=\langle a,b,s,t|[a,b]=1,\,a^s=(ab)^2,\,b^t=(ab)^2\rangle.

The morphism {a\mapsto a^2}, {b\mapsto b^2}, {s\mapsto s}, {t\mapsto t} is onto but not injective. Nevertheless, this group is {CAT(0)}.

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, {|det([\gamma_1],[\gamma_2])|}.

An immersed wall in a connected 2-dimensional complex is a graph that maps {\pi_1}-injectively to {X} and which lifts to a 2-sided embedding in {\tilde X}.

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 {G} acts freely on a {CAT(0)} 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 {\Lambda_i}, it is decidable wether the corresponding dual {CAT(0)} cube complex is finite dimensional.

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 {G} acts freely on a finite dimensional {CAT(0)} cube complex iff {G} is virtually special.

Such groups embed is {Sl(n,{\mathbb Z})}.

The proof consists in a thorough understanding of what the cube complex looks like, and an equivariant map to tree{\times{\mathbb R}^d}.

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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