Convergence group

In mathematics, a convergence group or a discrete convergence group is a group [math]\displaystyle{ \Gamma }[/math] acting by homeomorphisms on a compact metrizable space [math]\displaystyle{ M }[/math] in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary [math]\displaystyle{ \mathbb S^2 }[/math] of the hyperbolic 3-space [math]\displaystyle{ \mathbb H^3 }[/math]. The notion of a convergence group was introduced by Gehring and Martin (1987) and has since found wide applications in geometric topology, quasiconformal analysis, and geometric group theory.

Formal definition

Let [math]\displaystyle{ \Gamma }[/math] be a group acting by homeomorphisms on a compact metrizable space [math]\displaystyle{ M }[/math]. This action is called a convergence action or a discrete convergence action (and then [math]\displaystyle{ \Gamma }[/math] is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements [math]\displaystyle{ \gamma_n \in \Gamma }[/math] there exist a subsequence [math]\displaystyle{ \gamma_{n_k}, k=1,2,\dots }[/math] and points [math]\displaystyle{ a,b\in M }[/math] such that the maps [math]\displaystyle{ \gamma_{n_k}\big|_{M\setminus\{a\}} }[/math] converge uniformly on compact subsets to the constant map sending [math]\displaystyle{ M\setminus\{a\} }[/math] to [math]\displaystyle{ b }[/math]. Here converging uniformly on compact subsets means that for every open neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ b }[/math] in [math]\displaystyle{ M }[/math] and every compact [math]\displaystyle{ K\subset M\setminus \{a\} }[/math] there exists an index [math]\displaystyle{ k_0\ge 1 }[/math] such that for every [math]\displaystyle{ k\ge k_0, }[/math] [math]\displaystyle{ \gamma_{n_k}(K)\subseteq U }[/math]. Note that the "poles" [math]\displaystyle{ a, b\in M }[/math] associated with the subsequence [math]\displaystyle{ \gamma_{n_k} }[/math] are not required to be distinct.

Reformulation in terms of the action on distinct triples

The above definition of convergence group admits a useful equivalent reformulation in terms of the action of [math]\displaystyle{ \Gamma }[/math] on the "space of distinct triples" of [math]\displaystyle{ M }[/math]. For a set [math]\displaystyle{ M }[/math] denote [math]\displaystyle{ \Theta(M):=M^3\setminus \Delta(M) }[/math], where [math]\displaystyle{ \Delta(M)=\{(a,b,c)\in M^3\mid \#\{a,b,c\}\le 2\} }[/math]. The set [math]\displaystyle{ \Theta(M) }[/math] is called the "space of distinct triples" for [math]\displaystyle{ M }[/math].

Then the following equivalence is known to hold:

Let [math]\displaystyle{ \Gamma }[/math] be a group acting by homeomorphisms on a compact metrizable space [math]\displaystyle{ M }[/math] with at least two points. Then this action is a discrete convergence action if and only if the induced action of [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ \Theta(M) }[/math] is properly discontinuous.

Examples

• The action of a Kleinian group [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ \mathbb S^2=\partial \mathbb H^3 }[/math] by Möbius transformations is a convergence group action.
• The action of a word-hyperbolic group [math]\displaystyle{ G }[/math] by translations on its ideal boundary [math]\displaystyle{ \partial G }[/math] is a convergence group action.
• The action of a relatively hyperbolic group [math]\displaystyle{ G }[/math] by translations on its Bowditch boundary [math]\displaystyle{ \partial G }[/math] is a convergence group action.
• Let [math]\displaystyle{ X }[/math] be a proper geodesic Gromov-hyperbolic metric space and let [math]\displaystyle{ \Gamma }[/math] be a group acting properly discontinuously by isometries on [math]\displaystyle{ X }[/math]. Then the corresponding boundary action of [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ \partial X }[/math] is a discrete convergence action (Lemma 2.11 of ).

Classification of elements in convergence groups

Let [math]\displaystyle{ \Gamma }[/math] be a group acting by homeomorphisms on a compact metrizable space [math]\displaystyle{ M }[/math]with at least three points, and let [math]\displaystyle{ \gamma\in\Gamma }[/math]. Then it is known (Lemma 3.1 in or Lemma 6.2 in ) that exactly one of the following occurs:

(1) The element [math]\displaystyle{ \gamma }[/math] has finite order in [math]\displaystyle{ \Gamma }[/math]; in this case [math]\displaystyle{ \gamma }[/math] is called elliptic.

(2) The element [math]\displaystyle{ \gamma }[/math] has infinite order in [math]\displaystyle{ \Gamma }[/math] and the fixed set [math]\displaystyle{ \operatorname{Fix}_M(\gamma) }[/math] is a single point; in this case [math]\displaystyle{ \gamma }[/math] is called parabolic.

(3) The element [math]\displaystyle{ \gamma }[/math] has infinite order in [math]\displaystyle{ \Gamma }[/math] and the fixed set [math]\displaystyle{ \operatorname{Fix}_M(\gamma) }[/math] consists of two distinct points; in this case [math]\displaystyle{ \gamma }[/math] is called loxodromic.

Moreover, for every [math]\displaystyle{ p\ne 0 }[/math] the elements [math]\displaystyle{ \gamma }[/math] and [math]\displaystyle{ \gamma^p }[/math]have the same type. Also in cases (2) and (3) [math]\displaystyle{ \operatorname{Fix}_M(\gamma) = \operatorname{Fix}_M(\gamma^p) }[/math] (where [math]\displaystyle{ p\ne 0 }[/math]) and the group [math]\displaystyle{ \langle \gamma\rangle }[/math] acts properly discontinuously on [math]\displaystyle{ M\setminus \operatorname{Fix}_M(\gamma) }[/math]. Additionally, if [math]\displaystyle{ \gamma }[/math] is loxodromic, then [math]\displaystyle{ \langle \gamma\rangle }[/math] acts properly discontinuously and cocompactly on [math]\displaystyle{ M\setminus \operatorname{Fix}_M(\gamma) }[/math].

If [math]\displaystyle{ \gamma\in \Gamma }[/math] is parabolic with a fixed point [math]\displaystyle{ a\in M }[/math] then for every [math]\displaystyle{ x\in M }[/math] one has [math]\displaystyle{ \lim_{n\to\infty}\gamma^nx=\lim_{n\to-\infty}\gamma^nx =a }[/math] If [math]\displaystyle{ \gamma\in \Gamma }[/math] is loxodromic, then [math]\displaystyle{ \operatorname{Fix}_M(\gamma) }[/math] can be written as [math]\displaystyle{ \operatorname{Fix}_M(\gamma)=\{a_-,a_+\} }[/math] so that for every [math]\displaystyle{ x \in M\setminus \{a_-\} }[/math] one has [math]\displaystyle{ \lim_{n\to\infty}\gamma^nx=a_+ }[/math] and for every [math]\displaystyle{ x \in M\setminus \{a_+\} }[/math] one has [math]\displaystyle{ \lim_{n\to-\infty}\gamma^nx=a_- }[/math], and these convergences are uniform on compact subsets of [math]\displaystyle{ M\setminus \{a_-, a_+\} }[/math].

Uniform convergence groups

A discrete convergence action of a group [math]\displaystyle{ \Gamma }[/math] on a compact metrizable space [math]\displaystyle{ M }[/math] is called uniform (in which case [math]\displaystyle{ \Gamma }[/math] is called a uniform convergence group) if the action of [math]\displaystyle{ \Gamma }[/math] on [math]\displaystyle{ \Theta(M) }[/math] is co-compact. Thus [math]\displaystyle{ \Gamma }[/math] is a uniform convergence group if and only if its action on [math]\displaystyle{ \Theta(M) }[/math] is both properly discontinuous and co-compact.

Conical limit points

Let [math]\displaystyle{ \Gamma }[/math] act on a compact metrizable space [math]\displaystyle{ M }[/math] as a discrete convergence group. A point [math]\displaystyle{ x\in M }[/math] is called a conical limit point (sometimes also called a radial limit point or a point of approximation) if there exist an infinite sequence of distinct elements [math]\displaystyle{ \gamma_n\in \Gamma }[/math] and distinct points [math]\displaystyle{ a,b\in M }[/math] such that [math]\displaystyle{ \lim_{n\to\infty}\gamma_n x=a }[/math] and for every [math]\displaystyle{ y\in M\setminus \{x\} }[/math] one has [math]\displaystyle{ \lim_{n\to\infty}\gamma_n y=b }[/math].

An important result of Tukia, also independently obtained by Bowditch, states:

A discrete convergence group action of a group [math]\displaystyle{ \Gamma }[/math] on a compact metrizable space [math]\displaystyle{ M }[/math] is uniform if and only if every non-isolated point of [math]\displaystyle{ M }[/math] is a conical limit point.

Word-hyperbolic groups and their boundaries

It was already observed by Gromov that the natural action by translations of a word-hyperbolic group [math]\displaystyle{ G }[/math] on its boundary [math]\displaystyle{ \partial G }[/math] is a uniform convergence action (see for a formal proof). Bowditch proved an important converse, thus obtaining a topological characterization of word-hyperbolic groups:

Theorem. Let [math]\displaystyle{ G }[/math] act as a discrete uniform convergence group on a compact metrizable space [math]\displaystyle{ M }[/math] with no isolated points. Then the group [math]\displaystyle{ G }[/math] is word-hyperbolic and there exists a [math]\displaystyle{ G }[/math]-equivariant homeomorphism [math]\displaystyle{ M\to \partial G }[/math].

Convergence actions on the circle

An isometric action of a group [math]\displaystyle{ G }[/math] on the hyperbolic plane [math]\displaystyle{ \mathbb H^2 }[/math] is called geometric if this action is properly discontinuous and cocompact. Every geometric action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ \mathbb H^2 }[/math] induces a uniform convergence action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ \mathbb S^1 =\partial H^2\approx \partial G }[/math]. An important result of Tukia (1986), Gabai (1992), Casson–Jungreis (1994), and Freden (1995) shows that the converse also holds:

Theorem. If [math]\displaystyle{ G }[/math] is a group acting as a discrete uniform convergence group on [math]\displaystyle{ \mathbb S^1 }[/math] then this action is topologically conjugate to an action induced by a geometric action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ \mathbb H^2 }[/math] by isometries.

Note that whenever [math]\displaystyle{ G }[/math] acts geometrically on [math]\displaystyle{ \mathbb H^2 }[/math], the group [math]\displaystyle{ G }[/math] is virtually a hyperbolic surface group, that is, [math]\displaystyle{ G }[/math] contains a finite index subgroup isomorphic to the fundamental group of a closed hyperbolic surface.

Convergence actions on the 2-sphere

One of the equivalent reformulations of Cannon's conjecture, originally posed by James W. Cannon in terms of word-hyperbolic groups with boundaries homeomorphic to [math]\displaystyle{ \mathbb S^2 }[/math], says that if [math]\displaystyle{ G }[/math] is a group acting as a discrete uniform convergence group on [math]\displaystyle{ \mathbb S^2 }[/math] then this action is topologically conjugate to an action induced by a geometric action of [math]\displaystyle{ G }[/math] on [math]\displaystyle{ \mathbb H^3 }[/math] by isometries. This conjecture still remains open.

Applications and further generalizations

• Yaman gave a characterization of relatively hyperbolic groups in terms of convergence actions, generalizing Bowditch's characterization of word-hyperbolic groups as uniform convergence groups.
• One can consider more general versions of group actions with "convergence property" without the discreteness assumption.
• The most general version of the notion of Cannon–Thurston map, originally defined in the context of Kleinian and word-hyperbolic groups, can be defined and studied in the context of setting of convergence groups.