Let be a group acting by homeomorphisms on a compact metrizable space. This action is called a convergence action or a discrete convergence action if for every infinite distinct sequence of elements there exist a subsequence and points such that the maps converge uniformly on compact subsets to the constant map sending to. Here converging uniformly on compact subsets means that for every open neighborhood of in and every compact there exists an index such that for every . Note that the "poles" associated with the subsequence 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 on the "space of distinct triples" of. For a set denote, where. The set is called the "space of distinct triples" for. Then the following equivalence is known to hold: Let be a group acting by homeomorphisms on a compact metrizable space with at least two points. Then this action is a discrete convergence action if and only if the induced action of on is properly discontinuous.
Examples
The action of a Kleinian group on by Möbius transformations is a convergence group action.
The action of a word-hyperbolic group by translations on its ideal boundary is a convergence group action.
Let be a group acting by homeomorphisms on a compact metrizable space with at least three points, and let. Then it is known that exactly one of the following occurs: The element has finite order in ; in this case is called elliptic. The element has infinite order in and the fixed set is a single point; in this case is called parabolic. The element has infinite order in and the fixed set consists of two distinct points; in this case is called loxodromic. Moreover, for every the elements and have the same type. Also in cases and and the group acts properly discontinuously on. Additionally, if is loxodromic, then acts properly discontinuously and cocompactly on. If is parabolic with a fixed point then for every one has If is loxodromic, then can be written as so that for every one has and for every one has, and these convergences are uniform on compact subsets of.
A discrete convergence action of a group on a compact metrizable space is called uniform if the action of on is co-compact. Thus is a uniform convergence group if and only if its action on is both properly discontinuous and co-compact.
Conical limit points
Let act on a compact metrizable space as a discrete convergence group. A point is called a conical limit point if there exist an infinite sequence of distinct elements and distinct points such that and for every one has. An important result of Tukia, also independently obtained by Bowditch, states: A discrete convergence group action of a group on a compact metrizable space is uniform if and only if every non-isolated point of 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 on its boundary is a uniform convergence action. Bowditch proved an important converse, thus obtaining a topological characterization of word-hyperbolic groups: Theorem. Let act as a discrete uniform convergence group on a compact metrizable space with no isolated points. Then the group is word-hyperbolic and there exists a -equivariant homeomorphism.
An isometric action of a group on the hyperbolic plane is called geometric if this action is properly discontinuous and cocompact. Every geometric action of on induces a uniform convergence action of on. An important result of Tukia, Gabai, Casson–Jungreis, and Freden shows that the converse also holds: Theorem. If is a group acting as a discrete uniform convergence group on then this action is topologically conjugate to an action induced by a geometric action of on by isometries. Note that whenever acts geometrically on, the group is virtually a hyperbolic surface group, that is, 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, says that if is a group acting as a discrete uniform convergence group on then this action is topologically conjugate to an action induced by a geometric action of on 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.
