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 $k_{0}\geq 1$ such that for every $k\geq k_{0},$ $\gamma _{n_{k}}(K)\subseteq U$. 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 $\Delta (M)=\{(a,b,c)\in M^{3}\mid \#\{a,b,c\}\leq 2\}$. 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 inducted 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 $p\neq 0$ 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 $p\neq 0$) 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 $\lim _{n\to \infty }\gamma ^{n}x=\lim _{n\to -\infty }\gamma ^{n}x=a$ 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 $\lim _{n\to \infty }\gamma ^{n}x=a_{+}$ and for every $x\in M\setminus \{a_{+}\}$ one has $\lim _{n\to -\infty }\gamma ^{n}x=a_{-}$, 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.