Convergence group

In mathematics, a convergence group or a discrete convergence group is a group ${\displaystyle \Gamma }$ acting by homeomorphisms on a compact metrizable space ${\displaystyle M}$ in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary ${\displaystyle \mathbb {S} ^{2}}$ of the hyperbolic 3-space ${\displaystyle \mathbb {H} ^{3}}$. The notion of a convergence group was introduced by Gehring and Martin (1987) ^[1] and has since found wide applications in geometric topology, quasiconformal analysis, and geometric group theory.

Formal definition

Let ${\displaystyle \Gamma }$ be a group acting by homeomorphisms on a compact metrizable space ${\displaystyle M}$. This action is called a convergence action or a discrete convergence action (and then ${\displaystyle \Gamma }$ is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements ${\displaystyle \gamma _{n}\in \Gamma }$ there exist a subsequence ${\displaystyle \gamma _{n_{k}},k=1,2,\dots }$ and points ${\displaystyle a,b\in M}$ such that the maps ${\displaystyle \gamma _{n_{k}}{\big |}_{M\setminus \{a\}}}$ converge uniformly on compact subsets to the constant map sending ${\displaystyle M\setminus \{a\}}$ to ${\displaystyle b}$. Here converging uniformly on compact subsets means that for every open neighborhood ${\displaystyle U}$ of ${\displaystyle b}$ in ${\displaystyle M}$ and every compact ${\displaystyle K\subset M\setminus \{a\}}$ there exists an index ${\displaystyle k_{0}\geq 1}$ such that for every ${\displaystyle k\geq k_{0},}$ ${\displaystyle \gamma _{n_{k}}(K)\subseteq U}$. Note that the "poles" ${\displaystyle a,b\in M}$ associated with the subsequence ${\displaystyle \gamma _{n_{k}}}$ 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 ${\displaystyle \Gamma }$ on the "space of distinct triples" of ${\displaystyle M}$. For a set ${\displaystyle M}$ denote ${\displaystyle \Theta (M):=M^{3}\setminus \Delta (M)}$, where ${\displaystyle \Delta (M)=\{(a,b,c)\in M^{3}\mid \#\{a,b,c\}\leq 2\}}$. The set ${\displaystyle \Theta (M)}$ is called the "space of distinct triples" for ${\displaystyle M}$.

Then the following equivalence is known to hold:^[2]

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

Examples

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

Classification of elements in convergence groups

Let ${\displaystyle \Gamma }$ be a group acting by homeomorphisms on a compact metrizable space ${\displaystyle M}$with at least three points, and let ${\displaystyle \gamma \in \Gamma }$. Then it is known (Lemma 3.1 in ^[2] or Lemma 6.2 in ^[3]) that exactly one of the following occurs:

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

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

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

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

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

Uniform convergence groups

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

Conical limit points

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

An important result of Tukia,^[4] also independently obtained by Bowditch,^[2][5] states:

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

Word-hyperbolic groups and their boundaries

It was already observed by Gromov^[6] that the natural action by translations of a word-hyperbolic group ${\displaystyle G}$ on its boundary ${\displaystyle \partial G}$ is a uniform convergence action (see^[2] for a formal proof). Bowditch^[5] proved an important converse, thus obtaining a topological characterization of word-hyperbolic groups:

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

Convergence actions on the circle

An isometric action of a group ${\displaystyle G}$ on the hyperbolic plane ${\displaystyle \mathbb {H} ^{2}}$ is called geometric if this action is properly discontinuous and cocompact. Every geometric action of ${\displaystyle G}$ on ${\displaystyle \mathbb {H} ^{2}}$ induces a uniform convergence action of ${\displaystyle G}$ on ${\displaystyle \mathbb {S} ^{1}=\partial H^{2}\approx \partial G}$. An important result of Tukia (1986),^[7] Gabai (1992),^[8] Casson–Jungreis (1994),^[9] and Freden (1995)^[10] shows that the converse also holds:

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

Note that whenever ${\displaystyle G}$ acts geometrically on ${\displaystyle \mathbb {H} ^{2}}$, the group ${\displaystyle G}$ is virtually a hyperbolic surface group, that is, ${\displaystyle G}$ 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, (posed by James W. Cannon,^[11] although an earlier and more general conjecture, reducing to the Cannon conjecture for compact type, was given by Gaven J. Martin and Richard K. Skora ^[12])

These conjectures are in terms of word-hyperbolic groups with boundaries homeomorphic to ${\displaystyle \mathbb {S} ^{2}}$, says that if ${\displaystyle G}$ is a group acting as a discrete uniform convergence group on ${\displaystyle \mathbb {S} ^{2}}$ then this action is topologically conjugate to an action induced by a geometric action of ${\displaystyle G}$ on ${\displaystyle \mathbb {H} ^{3}}$ by isometries. These conjectures still remains open.

Applications and further generalizations

• Yaman gave a characterization of relatively hyperbolic groups in terms of convergence actions,^[13] 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.^[14]
• 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.^[15]