Well behaved subgroups of non-positively curved groups

Our main source for this post is Bridson and Haefliger’s book [BrH].

This post concerns contexts in which the subgroups of non-positively curved groups are well behaved. We have already considered hyperbolic groups; now we turn to the various classes of non–positively curved groups that allow zero-curvature.

${\sigma}$–quasi–convexity in semi–hyperbolic groups

We saw that quasi–convexity is a key notion when analysing subgroups of hyperbolic groups. Gersten and Short realised that quasi–convexity remains a key notion when studying subgroups of biautomatic groups. In [BrH] language–theoretic considerations are set aside and the results are presented for semi-hyperbolic groups, as we will here. (Similarly, Short carried the methods to the yet more general setting of bicombable groups.)

Recall that a subgroup ${H}$ of a group ${\Gamma}$ with a finite generating set ${A}$ is quasi–convex when there is some uniform constant ${k}$ such that geodesics in the Cayley graph ${C_A(\Gamma)}$ between pairs of points in ${H}$ stay in a ${k}$–neighbourhood of ${H}$.

The appropriate adaptation for a semi-hyperbolic group ${\Gamma}$ is to define quasi-convexity with respect to the semi-hyperbolic structure ${\{ \sigma_g \mid g \in \Gamma \}}$: a subgroup ${H}$ is ${\sigma}$–quasi–convex when there is some uniform constant ${k}$ such that for every pair ${g_1, g_2 \in H}$, the combing line ${\sigma_{{g_1}^{-1}g_2}}$ connecting them is in the ${k}$–neighbourhood of ${H}$.

Adapting the proofs for hyperbolic groups, one can prove:

Proposition. In a ${\sigma}$–semi–hyperbolic group ${\Gamma}$,

1. the intersection of any two ${\sigma}$–quasi–convex subgroups is again ${\sigma}$–quasi–convex;
2. centralisers of elements in ${\Gamma}$ are ${\sigma}$–quasi–convex.

The Algebraic Flat Torus Theorem ([BrH], pages 475–479 ). Suppose ${\Gamma}$ is a semi–hyperbolic group. If ${H}$ is a finitely generated abelian group then any monomorphism ${\phi: H \rightarrow \Gamma}$ is a quasi–isometric embedding.

The strategy is to argue, as we did for the proof of Theorem 2 in our last post, that the inclusions

$\displaystyle H \hookrightarrow Z(C(\phi(H))) \hookrightarrow C(\phi(H)) \hookrightarrow H$

are quasi-isometric embeddings and so the same is true of their composition.

The main application of the Algebraic Flat Torus Theorem is towards proving the “only if” part of:

Theorem ([BrH], page 479 ). A polycyclic group is a subgroup of a semi–hyperbolic group if and only if it is virtually abelian.

The idea is that a polycyclic subgroup that is not virtually abelian would give rise to a distorted abelian subgroup contrary to the Algebraic Flat Torus Theorem. One inducts on Hirsch length to reduce to the case ${H = \mathbb{Z}^n \rtimes_{\phi} \mathbb{Z}}$ where ${\phi \in \textup{GL}(n, \mathbb{Z})}$ and argues that the composition

$\displaystyle \mathbb{Z}^n \hookrightarrow \mathbb{Z}^n \rtimes_{\phi} \mathbb{Z} \hookrightarrow \Gamma$

cannot be a quasi–isometric embedding on account of the action of ${\mathbb{Z}}$ on ${ \mathbb{Z}^n}$ in the semi–direct product.

Similarly, we get:

Theorem. If ${p \neq q}$ then the Baumslag–Solitar group ${\langle a,b \mid b^{-1}a^p b = a^q \rangle}$ cannot be a subgroup of a semi–hyperbolic group.

Such a subgroup would give rise to an exponentially distorted ${{\mathbb Z}}$–subgroup, namely ${\langle a \rangle}$ — see Lemma 5 in our previous post.

Short pushed the notion of quasi–convexity to the more general context of bicombable groups, and established similar results such as:

Theorem (Short). Nilpotent subgroups of a bicombable group are virtually abelian.

[A group is virtually X when it has a finite index subgroup that is X.]

CAT(0) groups

Our account so far does not respect the historical development of the subject. The Algebraic’s Flat Torus Theory has an antecedent known as the Flat Torus Theorem of Gromoll–Wolf (Bull. Amer. Math. Soc., 77, 545–552, 1971) and, independently, Lawson–Yau (J. Diff. Geom., 7, 211–228, 1972). It concerns groups acting semi–simply by isometries on CAT(0) spaces (in particular, CAT(0) groups) and is of a similar character to the algebraic version, but is more technical to state. We will just give the key application:

Theorem. Every virtually solvable subgroup of a ${\textup{CAT}(0)}$–group is finitely generated and virtually abelian.

When is a subgroup of an automatic group automatic? When is a subgroup of a semi–hyperbolic group semi–hyperbolic?

Theorem 1 in our last post was that quasi–convex (or equivalently quasi–isometrically embedded) subgroups of hyperbolic groups are themselves hyperbolic. It is natural to ask: does a similar result hold for automatic groups or for semi–hyperbolic groups?

For automatic groups, the answer is “yes” —

Theorem (Gersten and Short). If ${\Gamma}$ is an automatic (or asynchronously automatic, or biautomatic) group via a combing ${\{ \sigma_g \mid g \in \Gamma \}}$ and ${H}$ is a ${\sigma}$–quasi–convex subgroup, then ${H}$ is automatic (or asynchronously automatic, or biautomatic).

In the case of semi-hyperbolic groups the answer appears to be a resounding “no”! —

Theorem (Bridson). There is a CAT(0)–group (so a semi–hyperbolic group) with a quasi–isometrically embedded subgroup that satisfies a slew of other nice properties and yet fails to be semi–hyperbolic itself.

Future directions

The theorem about solvable subgroups of CAT(0) groups goes further than the corresponding result for semi–hyperbolic groups, which only concerns polycyclic subgroups and Baumslag–Solitar subgroups. In the biautomatic setteing, the obstacle responsible for the shortfall is identified by Gersten and Short as:

Open question. Is every abelian subgroup of a biautomatic group finitely generated?

Presumably, the same question is open and represents the same obstacle for semi–hyperbolic groups.

What about the subgroups of automatic groups? Subgroups of biautomatic groups are implicitly discussed above in the more general context of semi–hyperbolic groups. Without the “bi-” it seems much less is known. [The two questions below due to Gersten can be found in his Problems on automatic groups, Algorithms and classification in combinatorial group theory, 225–232, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992 and also in the World of Groups problem list.]

Open question (Gersten). Is every retract of an automatic group automatic?

In his guided tour, Farb draws attention to a special case:

Open question. If ${G \times H}$ is automatic, is ${G}$ automatic?

The motivation for our final question is that it has a negative answer in the biautomatic (or, more generally, semi–hyperbolic) setting.

Open question (Gersten). Can ${\langle a, b \mid b^{-1}ab = a^2 \rangle}$ be a subgroup of an automatic group?

Update, 27 Feb, 2011 — added Gersten and Short’s Theorem about subgroups of automatic groups.

I have a question about the theorem of Bridson that provides a wild qi-embedded subgroup of a semi-hyperbolic group. Here, there seems to be an implicit assumption that qi-embedded subgroups are $\sigma$-quasi-convex. (The converse is obvious.) Is that the case?
I (Tim Riley) tried to avoid presenting that as an implicit assumption… but it is a niggling issue. It would seem unlikely that they are the same given that being qi-embedded makes no reference to ${\sigma}$. But finding an example may be tricky — distinguishing finer details of non–positive–curvature in groups often seems to be challenging.