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 edit

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.

• 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]).

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 ${\displaystyle M\setminus \{a_{-},a_{+}\}}$ .

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 edit

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 edit

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 ${\displaystyle M\to \partial G}$ .

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.

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 ${\displaystyle \mathbb {S} ^{2}}$ ,^[11] 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. This conjecture still remains open.

• Yaman gave a characterization of relatively hyperbolic groups in terms of convergence actions,^[12] 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.^[13]
• 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.^[14]