Relatively hyperbolic groups

Hyperbolic groups have all sorts of nice properties—they have linear isoperimetric inequalities, solvable word problem and conjugacy problem, are biautomatic, and so on. Two prime motivating examples of hyperbolic groups are fundamental groups of closed hyperbolic manifolds on the one hand, and free groups on the other.

Relaxing this definition a little, we remark that fundamental groups of cusped hyperbolic manifolds should still have many of the properties of hyperbolic groups, at least away from the cusps, where the characteristic properties of negative curvature break down a little.

This motivates (one) definition of a relatively hyperbolic group: a group G is hyperbolic relative to a collection of subgroups $\{H_1, \dots, H_k\}$ if G acts (geometrically) on some hyperbolic metric space X s.t. the quotient $X / G$ is quasi-isometric to the union of k copies of $[0,\infty)$ joined at 0.

Intuitively: each of the peripheral subgroups $H_k$ corresponds to a cusp, or some region where hyperbolicity breaks down; under a quasi-isometry which sends the compact core of the manifold to a point, each cusp (each of these “bad regions”) should be quasi-isometric to a ray going out to infinity.

Various definitions

This definition was formulated by Gromov in his seminal 1987 monograph on hyperbolic groups. There are many other definitions, coming from different motivations, many of which are equivalent. We describe them briefly here, dropping many technical adjectives (for full, careful statements, see e.g. section 3 of Hruska’s paper linked just above.)

Dynamical reformulations

First there is a slight reformulation of this by Bowditch—G is hyperbolic relative to its maximal parabolic subgroups* $\mathbb{P}$ if it acts on a hyperbolic metric space X, and the action is co-compact away from some equivariant collection of horoballs (in X) centered at the parabolic points of G.

*(Technically there can be infinitely many of these, but for the purposes of the definition, and for arguments, it suffices to take a set of representatives of conjugacy classes of maximal parabolics, which is [more often] finite.)

“Parabolic” here is defined in dynamical terms. We start with a dynamical axiomatization of Kleinian group actions on their limit sets: a (nonelementary) convergence group action is an action of a group G on a compact metric space with at least three points* s.t. the induced action of G on the space of distinct triples is properly discontinuous.

*(there are also elementary convergence group actions, which are the analogous objects when $|M| \leq 2$ , but we omit them here in the interest of brevity; see e.g. Hruska.)

Given a convergence group action, a loxodromic element is one which has infinite order and exactly two fixed points in M, and a subgroup $P \leq G$ is parabolic if it is infinite and contains no loxodromic element. A parabolic subgroup P has a unique fixed point $p \in M$ , which we call a parabolic point; stabilizers of parabolic points are maximal parabolic subgroups. A parabolic point p is bounded if its stabilizer acts cocompactly on $M - \{p\}$ .

Finally, a point $\xi \in M$ is a conical limit point if it exhibits a sort of generalized north-south dynamics (again, for the exact formulation, see e.g. Hruska), and a convergence group action is geometrically finite if every point of M is ether a conical limit point or a bounded parabolic point.

If that was too many definitions in a row: think of the case of Kleinian groups acting on their limit sets. To see that this axiomatization is a useful generalisation, we may point to e.g. a result of Tukia that shows every properly discontinuous action of a group G on a proper hyperbolic metric space induces a convergence group action on the boundary at infinity.

The theory of geometrically finite convergence group actions allows us to make another definition of relatively hyperbolic subgroups, proposed by Bowditch and worked out by Yaman: G is hyperbolic relative to a collection of subgroups $\mathbb{P}$ if it admits a geometrically finite convergence group action on a compact metric space M, with $\mathbb{P}$ as [a set of representatives of conjugacy classes of] maximal parabolics.

In fact, we can take M to be (i.e. M is G-equivariantly homeomorphic to) $\partial_\infty X$ for some hyperbolic metric space X on which G acts on properly; even more specifically, we may take that X to be the Cayley graph of G with combinatorial horoballs attached over the peripheral cosets (= cosets of the peripheral subgroups.)

These combinatorial horoballs are graphs (or in some cases 2-dimensional simplicial complexes) whose natural simplicial metrics combinatorially mimic the geometry of actual horoballs in negatively-curved spaces.

In one construction, formulated by Bowditch, a combinatorial horoball over $\Gamma$ is the graph with vertex set $P \times \mathbb{Z}_{\geq 0}$ and the “obvious” horizontal and vertical edges. The vertical edges are all assigned length 1, whereas the horizontal edges at level $P \times \{n\}$ are assigned length $2^{-n}$ . This has the effect of making the most efficient path between two points distance n apart in the same peripheral coset a horizontal path at level $\sim \log n$ , bookended by vertical ascent to / descent from that level.

A slightly different construction is given by Groves and Manning: their combinatorial horoballs have the same vertex set and vertical edges, but the horizontal edges at level k are different: such an edge exists between any $(v,k)$ and $(w,k)$ whenever $0 < d_\Gamma(v,w) \leq 2^k$ , and all of these edges have length 1. There are also (horizontal and vertical) 2-cells attached, although these are ignored when regarding the combinatorial horoball as a metric space.

This last definition might be thought of a dynamical reformulation / abstraction / deconstruction of Gromov’s original definition.

Electrifying and fine alternatives

A different definition was proposed by Farb for finitely-generated groups, and generalised to non-f.g. groups by Bowditch and Osin: a group G is hyperbolic relative to a collection of subgroups $\mathbb{P} = \{P_1, \dots, P_n\}$ iff the electrified Cayley graph, formed by taking a Cayley graph by adding to the Cayley graph a vertex (“cone point”) for each left coset $gP_i$ and edges of length 1/2 from this new vertex to each element of $gP_i$ , is Gromov-hyperbolic, and exhibits Bounded Coset Penetration (BCP).

This definition is motivated more directly by the structure of a free product acting on its Bass-Serre tree, where geodesics pass through vertices in very controlled ways.

The BCP condition aims to mimic this, in a quasi sense: in short (the full statement is rather technical), it gives control, up to bounded error, over quasi-geodesics which penetrate (pass through) cosets (cusp neighborhoods.) Given such a quasi-geodesic, call the vertex immediately preceding the cone point the entering vertex, and the one immediately following the cone point the exiting vertex. BCP then stipulates that for any two quasigeodesics $\gamma, \gamma'$ which start and end at (essentially) the same point,

if $\gamma$ and $\gamma'$ penetrate a coset gP, then the entering vertices of $\gamma$ and $\gamma'$ are bounded distance apart in the [unelectrified] Cayley graph (with bound depending only on the quasi-geodesic constants), as are the exiting vertices;
if $\gamma$ penetrates gP but not $\gamma'$ does not, then the entering and exiting vertices of $\gamma$ are bounded distance apart.

(In the language of cusps: if the quasi-geodesics both go through a cusp neighborhood, then they stay close near where they enter and exit; if one goes through a cusp, but not the other, then the former cannot stay in the cusp for very long.)

An abstraction of Farb’s approach was proposed and explored by Bowditch: call a graph is called fine if each of its edges is contained in finitely many cycles of length n for each n. Fine graphs capture, in graph-theoretic terms, the BCP, although their equivalence can be, not to put too fine a point on it, subtle.

Fine graphs give us the following (fifth) definition of relative hyperbolicity: G is hyperbolic relative to a collection of subgroups $\mathbb{P}$ if it acts acts (properly discontinuously and co-compactly) on a fine Gromov-hyperbolic graph, with $\mathbb{P}$ a set of representatives of the conjugacy classes of infinite vertex stabilizers.

Bowditch gives an explicit construction of this graph: starting with a hyperbolic space on which G acts (e.g. the Cayley graph augmented with combinatorial horoballs, as described above), and form a graph K with a vertex for each horoball (of fixed level t), and an edge between two vertices if the corresponding horoballs are $\leq 2t$ apart.

Via relative Dehn functions

Osin gives a different (sixth) definition in terms of relative Dehn (isoperimetric) functions: G is hyperbolic relative to $\mathbb{P} = \{P_1, \dots, P_n\}$ if it has a finite relative presentation, and the relative Dehn function of G is well-defined and linear for some (and hence every) finite relative presentation.

Here a relative presentation is a set S which together with the peripheral subgroups generates G and a set of “relators” whose normal closure K is the kernel of $F(S) * (* \mathcal{P}) \to G$ , and a relative Dehn function is a Dehn function for the relative presentation, with conjugating elements for the relators taken from K (for a less terse / cryptic definition, again see e.g. Hruska, or Osin.)

The motivating model geometry (according to Hruska—I’m not sure I see it at the moment) is apparently still essentially that of a free product acting on its Bass-Serre tree.

Basically a tree grading

Considering what geodesics in a relatively hyperbolic group can look like—they essentially run from cusp (peripheral coset) to cusp (peripheral coset) along more-or-less hyperbolic geodesics (see below)—motivates (or, actually, yields, after some proof) a different (seventh!) definition, given by Druțu-Sapir: a group G is hyperbolic relative to $\mathbb{P} = \{P_1, \dots, P_n\}$ if all of its asymptotic cones are tree-graded, with the pieces being left cosets of the $\mathbb{P}$ .

Or, less precisely but without using the words “asymptotic cone”: relatively hyperbolic groups look coarsely like tree-graded spaces, with the peripheral cosets being the pieces.

One may note the similarities between this picture and the geometry of a CAT(0) space with isolated flats, and indeed these were an important motivation for Druțu-Sapir.

Equivalence of notions

All of the above definitions are equivalent for finitely-generated groups, and almost all of them—except the tree-graded one, whose definition requires finite generation—are equivalent for countable groups.

Nevertheless, as noted above, they have disparate motivating origins, which hints at the possible richness of the theory and of techniques which can be applied to study relatively hyperbolic groups.

Examples

As pointed above, fundamental groups of punctured hyperbolic surfaces are prime motivating examples of relatively hyperbolic groups. These are hyperbolic relative to their cusp groups (which are all infinite cyclic groups.)

Free products, relative to their free factors, are another prime motivating example. Indeed Gromov originally described his formulation of relative hyperbolicity, in his landmark paper on hyperbolic groups, as “a hyperbolic version of small cancellation theory over free products by adopting geometric language of manifolds with cusps”.

CAT(0) groups acting on spaces with isolated flats are hyperbolic relative to the stabilizers of maximal flats are a third important class of examples, as noted above in the description of the Druțu-Sapir definition based on tree-graded spaces.

Behrstock-Hagen, building on work on Hruska-Kleiner, have a criterion, in terms of the simplicial boundary, for when cubulated groups are hyperbolic relative to specified families of subgroups. In particular, not all CAT(0) groups, or even all cubulated groups, can be relatively hyperbolic—e.g. right-angled Artin groups (RAAGs) are not.

The non-examples are just as important as the examples, in terms of pointing out what the theory is good for and what its limitations are. For instance, higher-rank free abelian groups (e.g. $\mathbb{Z}^2$ ) are not hyperbolic relative to any finite collection of their subgroups (e.g. $\{\mathbb{Z}\}$ )—the electrified Cayley graph here is hyperbolic, but not fine, i.e. the BCP is not satisfied. Indeed, in some sense, there are “too many bad regions” which are “not sufficiently separated”, and so the theory of relative hyperbolicity does not help here.

Mapping class groups are not hyperbolic relative to any collection of subgroups, by a result of Behrstock–Druțu–Mosher, except in sporadic cases where they are virtually free; the same authors used similar arguments to show that many other classes of groups, including outer automorphism groups of free groups, lattices in higher-rank Lie groups, and fundamental groups of graph manifolds, are not hyperbolic relative to any collection of their subgroups. The key notion in their arguments is that of thickness, which appears to capture, intuitively, the notion of flats or cusps clustering or interacting in a way which conspires against the slight weakening of negative curvature implied by relative hyperbolicity.

A different notion of weakened hyperbolicity, hierarchical hyperbolicity, inspired by the structure of the mapping class group as described by Masur-Minsky, does apply to many (though not all) of these examples: mapping class groups and RAAGs (indeed, all cubulated groups) are hierarchically hyperbolic, for instance.

Properties

Quite a few of these were formulated as “equivalent definitions” above, in particular,

linear relative Dehn function (indeed, relative hyperbolic groups have relative Dehn presentations, and hence e.g. solvable word and conjugacy problems, provided the word and conjugacy problems are solvable in the peripheral subgroups), and
tree grading of asymptotic cones, which yields information about geodesics—see below.

Which makes one wonder a little—where is the line, if there is any, between “definition” and “property”?

Relative hyperbolicity and hyperbolicity

A group which is relatively hyperbolic to a collection of peripheral subgroups, each of which is (word-)hyperbolic, is itself hyperbolic.

Conversely, hyperbolic groups are hyperbolic relative to collections of quasiconvex malnormal subgroups with bounded coset intersection—e.g. the fundamental group of a once-punctured torus (which is $\cong F_2 = \langle a, b \rangle$ ) is hyperbolic, but also hyperbolic relative to the cusp group $\langle [a,b] \rangle$ .

Broadly speaking, it seems possible to push through arguments that lead to analogues of properties of hyperbolic groups in many cases, but we don’t seem to have as many nice general results as we do in the hyperbolic case …

Quasiconvex subgroups and distortion

As an illustration of this broad principle: Hruska, in his paper linked to above, defined a notion of relatively quasiconvex subgroups of relatively hyperbolic groups, showed that these are also relatively hyperbolic, that the intersection of relatively quasiconvex subgroups is relatively quasiconvex, and that every undistorted subgroup of a finitely-generated relatively hyperbolic group is relatively quasiconvex.

Geodesics

What do geodesics here look like? The Druțu-Sapir definition provides some answers to this question: relatively hyperbolic groups are coarsely tree-graded, and so we can expect properties of geodesics in tree-graded spaces to hold coarsely for (Cayley graphs of) relatively hyperbolic groups, as described by Alex Sisto on his blog.

For instance: geodesics in tree-graded spaces are essentially unique, modulo what they do within the pieces; in a relatively hyperbolic group, this remains true (roughly speaking), up to some bounded error in where the geodesic enters and exits the pieces.

A more precise way of formulating this is in terms of projections $\pi_P$ to each of the pieces P: given any point x in our space, $\pi_P(x)$ is the unique point in $P$ which every geodesic from x to P must go through. With these projections defined, we can say even more. In a tree-graded space, if $\pi_P(x) \neq \pi_P(y)$ , then any geodesic between two points must go through P. The appropriately coarsified version of this is true for relatively hyperbolic groups:

If the images of two points x and y under projection to a peripheral coset P are at least C apart (where C is some constant that depends only on the group and choice of peripheral subgroups), then any geodesic between x and y goes from x to near $\pi_P(x)$ , tracks P to $\pi_P(y)$ , and then goes to y from there. The geodesic may track several peripheral subsets, in turn, this way, going between them in an essentially unique (hyperbolic) way; additional structural results, again analogous to results for tree-graded spaces, tell us more about the order in which these peripheral subsets appear, and so on.

Geodesic flows?

Gromov defined—and Champetier clarified his definition of—a geodesic flow on any word-hyperbolic group. Mineyev gave a more general construction, using a homological bicombing—roughly speaking, a homology analogue of a map which associates to each pair of group elements g, h a geodesic (segment) between them, extended also in a sensible way to the Gromov boundary. Mineyev’s construction, in fact, more generally produces a flow on any hyperbolic simplicial complex, so it may be possible to apply it more or less directly to obtain some sort of geodesic flow on a relatively hyperbolic group, or at any rate on its cusped space.

Groves-Manning, using their combinatorial horoballs (they call the result of attaching these to the Cayley graph “the cusped space”), construct an analogue of Mineyev’s homological bicombing for relatively hyperbolic groups. It may be possible to use this to obtain a geodesic flow on a relatively hyperbolic group, analogous to Mineyev’s original construction.

Both of these may in fact be possible, and one of them may have better / more natural properties—likely the latter (?): the former seems more naturally a flow on the cusped space, and may or may not descend in a reasonable way to the group / original Cayley graph.

(The existing literature does not seem to have anything explicitly / directly addressing either of these possibilities.)

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Relatively hyperbolic groups