Convergence group

In mathematics, a convergence group or a discrete convergence group is a group $$\Gamma$$ acting by homeomorphisms on a compact metrizable space $$M$$ in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary $$\mathbb S^2$$ of the hyperbolic 3-space $\mathbb H^3 $. 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 $$\Gamma$$ be a group acting by homeomorphisms on a compact metrizable space $$M$$. This action is called a convergence action or a discrete convergence action (and then $$\Gamma$$ is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements $$\gamma_n \in \Gamma$$ there exist a subsequence $$\gamma_{n_k}, k=1,2,\dots$$ and points $$a,b\in M$$ such that the maps $$\gamma_{n_k}\big|_{M\setminus\{a\}}$$ converge uniformly on compact subsets to the constant map sending $$M\setminus\{a\}$$ to $$b$$. Here converging uniformly on compact subsets means that for every open neighborhood $$U$$ of $$b$$ in $$M$$ and every compact $$ K\subset M\setminus \{a\}$$ there exists an index $$k_0\ge 1$$ such that for every $$k\ge k_0,$$ $$ \gamma_{n_k}(K)\subseteq U$$. Note that the "poles" $$ a, b\in M$$ associated with the subsequence $$\gamma_{n_k}$$ 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 $$\Gamma$$ on the "space of distinct triples" of $$M$$. For a set $$M$$ denote $$\Theta(M):=M^3\setminus \Delta(M)$$, where $$\Delta(M)=\{(a,b,c)\in M^3\mid \#\{a,b,c\}\le 2\}$$. The set $$\Theta(M)$$ is called the "space of distinct triples" for $$M$$.

Then the following equivalence is known to hold:

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

Examples

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

Classification of elements in convergence groups
Let $$\Gamma$$ be a group acting by homeomorphisms on a compact metrizable space $$M$$with at least three points, and let $$\gamma\in\Gamma$$. Then it is known (Lemma 3.1 in or Lemma 6.2 in ) that exactly one of the following occurs:

(1) The element $$\gamma$$ has finite order in $$\Gamma $$; in this case $$\gamma$$ is called elliptic.

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

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

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

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

Uniform convergence groups
A discrete convergence action of a group $$\Gamma$$ on a compact metrizable space $$M$$ is called uniform (in which case $$\Gamma$$ is called a uniform convergence group) if the action of $$\Gamma$$ on $$\Theta(M)$$ is co-compact. Thus $$\Gamma$$ is a uniform convergence group if and only if its action on $$\Theta(M)$$ is both properly discontinuous and co-compact.

Conical limit points
Let $$\Gamma$$ act on a compact metrizable space $$M$$ as a discrete convergence group. A point $$x\in M$$ 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 $$\gamma_n\in \Gamma$$ and distinct points $$a,b\in M$$ such that $$\lim_{n\to\infty}\gamma_n x=a$$ and for every $$y\in M\setminus \{x\}$$ one has $$ \lim_{n\to\infty}\gamma_n y=b$$.

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

A discrete convergence group action of a group $$\Gamma$$ on a compact metrizable space $$M$$ is uniform if and only if every non-isolated point of $$M$$ 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 $$G$$ on its boundary $$\partial G$$ 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 $$ G$$ act as a discrete uniform convergence group on a compact metrizable space $$M$$ with no isolated points. Then the group $$G$$ is word-hyperbolic and there exists a $$G$$-equivariant homeomorphism $$M\to \partial G$$.

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

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

Note that whenever $$G$$ acts geometrically on $$\mathbb H^2 $$, the group $$G$$ is virtually a hyperbolic surface group, that is,  $$G$$ 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 $$\mathbb S^2$$, says that if $$G$$ is a group acting as a discrete uniform convergence group on $$\mathbb S^2$$ then this action is topologically conjugate to an action induced by a geometric action of $$G$$ on $$\mathbb H^3$$ 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.