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 $\gamma_, k=1,2,\dots$ and points $a,b\in M$ such that the maps $\gamma_\big, _$ converge uniformly on compact subsets to the constant map sending $M\setminus\$ 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 \$ there exists an index $k_0\ge 1$ such that for every $k\ge k_0,$ $\gamma_(K)\subseteq U$. Note that the "poles" $a, b\in M$ associated with the subsequence $\gamma_$ 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)=\$. 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 $\operatorname_M(\gamma)$ is a single point; in this case $\gamma$ is called ''parabolic''.

(3) The element $\gamma$ has infinite order in $\Gamma$ and the fixed set $\operatorname_M(\gamma)$ 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) $\operatorname_M(\gamma) = \operatorname_M(\gamma^p)$ (where $p\ne 0$) and the group $\langle \gamma\rangle$ acts properly discontinuously on $M\setminus \operatorname_M(\gamma)$. Additionally, if $\gamma$ is loxodromic, then $\langle \gamma\rangle$ acts properly discontinuously and cocompactly on $M\setminus \operatorname_M(\gamma)$.

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

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 $\lim_\gamma_n x=a$ and for every $y\in M\setminus \$ one has $\lim_\gamma_n y=b$.

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.

References

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Group theory
Dynamical systems
Geometric topology
Geometric group theory