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RESEARCH ANNOUNCEMENT: THE STRUCTURE OF

GROUPS WITH A QUASICONVEX HIERARCHY

DANIEL T. WISE

Abstract. Let G be a word-hyperbolic group with a quasiconvex hier- archy. We show that G has a finite index subgroup G

0

that embeds as a

quasiconvex subgroup of a right-angled artin group. It follows that every

quasiconvex subgroup of G is a virtual retract, and is hence separable.

The results are applied to certain 3-manifold and one-relator groups.

1. Introduction and main results

This announcement concerns a growing body of work much of which is

joint with Fr ́ed ́eric Haglund, Chris Hruska, Tim Hsu, and Michah Sageev.

Many groups that arise naturally in topology and combinatorial group

theory (e.g. a one-relator group, or 3-manifold group, or HNN extension

of a free group along a cyclic subgroup) are associated with small low- dimensional objects. The overall picture presented here suggests that it

can be very fruitful to sacrifice the small initial “presentation” in favor of a

higher-dimensional but much more organized structure, since this can reveal

many hidden properties of the group.

1.1. Main theorem.

Definition 1.1 (Quasiconvex hierarchy). A trivial group has a length 0

quasiconvex hierarchy. For h ≥ 1, a group G has a length h quasiconvex

hierarchy if G ∼= A ∗C B or G ∼= A∗Ct=C0 where A, B have quasiconvex

hierarchies of length ≤ (h−1), and C is a finitely generated group such that

the map C → G is a quasi-isometry with respect to word metrics.

The main result is the following [Wis]:

Theorem 1.2. If G is word-hyperbolic and has a quasiconvex hierarchy then

G is the fundamental group of a compact nonpositively curved cube complex

X that is virtually special.

Date: October 16, 2009.

2000 Mathematics Subject Classification. 53C23, 20F36, 20F55, 20F67, 20F65, 20E26.

Key words and phrases. CAT(0) cube complex, right-angled artin group, subgroup

separable, 3-manifold, one-relator group.

Research supported by a grant from NSERC. This announcement was written while on

sabbatical at the Einstein Institute of Mathematics at the Hebrew University. I am grateful

to the Lady Davis Foundation for its support, and to Zlil Sela for helpful conversations.

1

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RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY2

A similar result holds for many groups that are hyperbolic relative to

virtually abelian subgroups, and I expect that it holds in general for such

groups. However, hyperbolicity cannot be relaxed too much here: For in- stance, the Baumslag-Solitar group ha, t | (a

2

)

t = a

3

i is an example of a

one-relator group with a (nonquasiconvex) hierarchy, and there are torsion- free irreducible lattices in Aut(T × T) that have quasiconvex hierarchies

[BM97, Wis07], but none of these groups is virtually special.

1.2. Application to one-relator groups. A one-relator group is a group

having a presentation ha, b, · · · | Wn

i with a single defining relation. As- suming that W is reduced and cyclically reduced and is not a proper power,

the one-relator group has torsion if and only if n ≥ 2. In this case, all

torsion is conjugate into hWi ∼= Zn and the group is virtually torsion-free.

We refer to [LS77] for more information on one-relator groups. A significant

feature of one-relator groups with torsion is that they are word-hyperbolic,

since the Newman Spelling Theorem provides very strong small-cancellation

behavior. It became clear in the 60’s that one-relator groups with torsion

are better behaved than general one-relator groups, and to test this Gilbert

Baumslag made the following:

Conjecture 1.3 ([Bau67]). Every one-relator group with torsion is residu- ally finite.

The main tool for studying one-relator groups is the Magnus hierarchy.

Roughly speaking, every one-relator group G is an HNN extension H∗Mt=M0

of a simpler one-relator group H where M and M0 are free subgroups gen- erated by subsets of the generators of the presentation of G. The hierarchy

terminates at a virtually free group of the form Zn ∗ F. For one-relator

groups with torsion, we show that the subgroups M, M0 are quasiconvex

at each level of the hierarchy in [Wis]. This result depends upon a variant

of the Newman spelling theorem [HW01, Lau07]. When G is a one-relator

group with torsion, and G0

is a torsion-free finite index subgroup, the in- duced hierarchy for G0

is a quasiconvex hierarchy that terminates at trivial

groups (instead of finite groups) and is thus covered by Theorem 1.2.

Theorem 1.4. Every one-relator group with torsion is virtually special.

As discussed in Section 3, a virtually special word-hyperbolic has very

strong properties, and in particular it is residually finite, so Conjecture 1.3

follows from Theorem 1.4.

1.3. Application to 3-manifolds. Prior to Thurston’s work, the main

tool used to study 3-manifolds is a hierarchy which is a sequence of splittings

along incompressible surfaces until only 3-balls remain. (An incompressible

surfaces is a 2-sided π1-injective surface along which the fundamental group

splits as either an HNN extension or amalgamated free product.)

It is well-known that every irreducible 3-manifold with an incompress- ible surface has a hierarchy and every irreducible 3-manifold with boundary

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RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY3

has an incompressible surface. It is a deeper result that for a finite volume

3-manifold with cusps, there is always an incompressible geometrically fi- nite surface [CS84]. In general, an incompressible surface in a hyperbolic

3-manifold is either geometrically finite or virtually corresponds to a fiber

(see [Bon86]). A fundamental result of Thurston’s about subgroups of fun- damental groups of infinite volume hyperbolic manifolds ensures that if the

initial incompressible surface is geometrically finite, then the further incom- pressible surfaces in (any) hierarchy are geometrically finite (see the survey

in [Can94]). Finally, we note that the geometrical finiteness of an incom- pressible surface where the 3-manifold splits corresponds precisely to the

quasi-isometric embedding of the corresponding subgroup along which the

fundamental group splits. Thus, if M has an incompressible surface then

π1M has a quasiconvex hierarchy and we have:

Theorem 1.5. If M is a hyperbolic 3-manifold with an incompressible ge- ometrically finite surface then π1M is virtually special.

We expect that all hyperbolic fibered 3-manifolds have finite covers with

incompressible geometrically finite surfaces, so that all Haken hyperbolic

3-manifolds are virtually special.

Corollary 1.6. If M is a hyperbolic 3-manifold with an incompressible ge- ometrically finite surface then π1M is subgroup separable.

In the 80’s Thurston suggested that perhaps every hyperbolic 3-manifold

is virtually fibered. The key to proving the virtual fibering problem is the

following beautiful result which weaves together several important ideas from

3-manifold topology [Ago08]:

Proposition 1.7 (Agol’s fibering criterion). Let M be a compact 3-manifold,

and suppose that π1M is residually finite Q-solvable. (This holds when π1M

is residually torsion-free nilpotent). Then M has a finite cover that fibers.

For a haken hyperbolic 3-manifold M, either it virtually fibers, or the

first incompressible surface is geometrically finite. In this case the virtual

specialness implies that M has a finite cover with π1Mc contained in a graph

group which is residually torsion-free nilpotent, so we have:

Corollary 1.8. Every hyperbolic haken 3-manifold is virtually fibered.

1.4. Application to limit groups. Fully residually free groups or limit

groups have been a recent focal point of geometric group theory. These are

groups G with the property that for every finite set g1, . . . , gk of nontrivial

elements, there is a free quotient G → G ̄ such that ̄g1, . . . , g ̄k are nontrivial.

Among the many wonderful properties proved for these groups is that they

have a rather simple cyclic hierarchy terminating at free groups.

(1) A ∗Z B where Z is cyclic and malnormal in A, and A, B have such

hierarchies.

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RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY4

(2) A∗Zt=Z0 where Z is cyclic and malnormal in A and Z, Z0 do not have

nontrivially intersecting conjugates.

(3) A ∗Z B where Z is cyclic and malnormal in A and B ∼= Z × Z

n

for

some n.

This hierarchy was obtained in [KM98], and is also implicit in Sela’s re- tractive tower description of limit groups [Sel03]. This hierarchy allows one

to prove that limit groups are hyperbolic relative to free abelian subgroups

[Dah03, Ali05]. Using this cyclic hierarchy and the relative hyperbolicity we

are able to see that:

Corollary 1.9. Every limit group is virtually special.

Combined with subgroup separability results for virtually special groups

that are hyperbolic relative to abelian subgroups we are able to recover

Wilton’s result that limit groups are subgroup separable [Wil08].

2. Cubulating Groups

2.1. Nonpositively curved cube complexes. Gromov introduced non- positively curved cube complexes as a source of simple examples, but they

have turned out to be playing an unexpectedly wide role. An n-cube is a

copy of [−1, 1]n and a 0-cube is a single point. We regard the boundary of

an n-cube as consisting of the union of lower dimensional cubes. A cube

complex is a cell complex formed from cubes, such that the attaching map

of each cube is combinatorial in the sense that it sends cubes homeomorphi- cally to cubes by a map modelled on a combinatorial isometry of n-cubes.

The link of a 0-cube v is the complex whose 0-simplices correspond to ends

of 1-cubes adjacent to v, and these 0-simplices are joined up by n-simplices

for each corner of an (n + 1)-cube adjacent to v.

A flag complex is a simplicial complex with the property that any finite

pairwise-adjacent collection of vertices spans a simplex. A cube complex C

is nonpositively curved if link(v) is a flag complex for each 0-cube v ∈ C

0

.

A map φ : Y → X between nonpositively curved cube complex is a local- isometry if for each y ∈ Y

0

the induced map link(y) → link(φ(y)) is an

adjacency preserving embedding.

2.2. CAT(0) cube complexes and hyperplanes. Simply-connected non- positively curved cube complexes are called CAT(0) cube complexes because

they admit a CAT(0) metric where each n-cube is isometric to [−1, 1]n ⊂ R

n

however we rarely use this metric. Instead, the crucial characteristic prop- erties of CAT(0) cube complexes are the separative qualities of their hy- perplanes: A midcube is the codimension-1 subspace of the n-cube [−1, 1]n

obtained by restricting exactly one coordinate to 0. A hyperplane is a con- nected nonempty subspace of C whose intersection with each cube is either

empty or consists of one of its midcubes. The 1-cells intersected by a hy- perplane are dual to it.

Remark 2.1. Hyperplanes have several important properties [Sag95]: