Convergence group

In mathematics, a convergence group or a discrete convergence group is a group formula_1 acting by homeomorphisms on a compact metrizable space formula_2 in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary formula_3 of the hyperbolic 3-space formula_4.
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 formula_1 be a group acting by homeomorphisms on a compact metrizable space formula_2. This action is called a "convergence action" or a "discrete convergence action" (and then formula_1 is called a "convergence group" or a "discrete convergence group" for this action) if for every infinite distinct sequence of elements formula_8 there exist a subsequence formula_9 and points formula_10 such that the maps formula_11 converge uniformly on compact subsets to the constant map sending formula_12 to formula_13. Here converging uniformly on compact subsets means that for every open neighborhood formula_14 of formula_13 in formula_2 and every compact formula_17 there exists an index formula_18 such that for every formula_19 formula_20. Note that the "poles" formula_21 associated with the subsequence formula_22 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 formula_1 on the "space of distinct triples" of formula_2.
For a set formula_2 denote formula_26, where formula_27. The set formula_28 is called the "space of distinct triples" for formula_2.
Then the following equivalence is known to hold:
Let formula_1 be a group acting by homeomorphisms on a compact metrizable space formula_2 with at least two points. Then this action is a discrete convergence action if and only if the induced action of formula_1 on formula_28 is properly discontinuous.
Classification of elements in convergence groups.
Let formula_1 be a group acting by homeomorphisms on a compact metrizable space formula_2with at least three points, and let formula_47. Then it is known (Lemma 3.1 in or Lemma 6.2 in ) that exactly one of the following occurs:
(1) The element formula_48 has finite order in formula_49; in this case formula_48 is called "elliptic".
(2) The element formula_48 has infinite order in formula_49 and the fixed set formula_53 is a single point; in this case formula_48 is called "parabolic".
(3) The element formula_48 has infinite order in formula_1 and the fixed set formula_53 consists of two distinct points; in this case formula_48 is called "loxodromic".
Moreover, for every formula_59 the elements formula_48 and formula_61have the same type. Also in cases (2) and (3) formula_62 (where formula_59) and the group formula_64 acts properly discontinuously on formula_65. Additionally, if formula_48 is loxodromic, then formula_64 acts properly discontinuously and cocompactly on formula_68.
If formula_69 is parabolic with a fixed point formula_70 then for every formula_71 one has formula_72
If formula_69 is loxodromic, then formula_53 can be written as formula_75 so that for every formula_76 one has formula_77 and for every formula_78 one has formula_79, and these convergences are uniform on compact subsets of formula_80.
Uniform convergence groups.
A discrete convergence action of a group formula_1 on a compact metrizable space formula_2 is called "uniform" (in which case formula_1 is called a "uniform convergence group") if the action of formula_1 on formula_28 is co-compact. Thus formula_1 is a uniform convergence group if and only if its action on formula_28 is both properly discontinuous and co-compact.
Conical limit points.
Let formula_1 act on a compact metrizable space formula_2 as a discrete convergence group. A point formula_71 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 formula_91 and distinct points formula_10 such that formula_93 and for every formula_94 one has formula_95.
An important result of Tukia, also independently obtained by Bowditch, states:
A discrete convergence group action of a group formula_1 on a compact metrizable space formula_2 is uniform if and only if every non-isolated point of formula_2 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 formula_36 on its boundary formula_37 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 formula_101 act as a discrete uniform convergence group on a compact metrizable space formula_2 with no isolated points. Then the group formula_36 is word-hyperbolic and there exists a formula_36-equivariant homeomorphism formula_105.
Convergence actions on the circle.
An isometric action of a group formula_36 on the hyperbolic plane formula_107 is called "geometric" if this action is properly discontinuous and cocompact. Every geometric action of formula_36 on formula_107 induces a uniform convergence action of formula_36 on formula_111.
An important result of Tukia (1986), Gabai (1992), Casson–Jungreis (1994), and Freden (1995) shows that the converse also holds:
Theorem. If formula_36 is a group acting as a discrete uniform convergence group on formula_113 then this action is topologically conjugate to an action induced by a geometric action of formula_36 on formula_115 by isometries.
Note that whenever formula_36 acts geometrically on formula_107, the group formula_36 is virtually a hyperbolic surface group, that is, formula_36 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 formula_3, says that if formula_36 is a group acting as a discrete uniform convergence group on formula_3 then this action is topologically conjugate to an action induced by a geometric action of formula_36 on formula_124 by isometries. This conjecture still remains open.