acting by homeomorphisms on a compact metrizable space
in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary
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.
be a group acting by homeomorphisms on a compact metrizable space
is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements
converge uniformly on compact subsets to the constant map sending
Here converging uniformly on compact subsets means that for every open neighborhood
The above definition of convergence group admits a useful equivalent reformulation in terms of the action of
is called the "space of distinct triples" for
be a group acting by homeomorphisms on a compact metrizable space
be a group acting by homeomorphisms on a compact metrizable space
consists of two distinct points; in this case
acts properly discontinuously and cocompactly on
, and these convergences are uniform on compact subsets of
A discrete convergence action of a group
is called a uniform convergence group) if the action of
is a uniform convergence group if and only if its action on
act on a compact metrizable space
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
An important result of Tukia,[4] also independently obtained by Bowditch,[2][5] states: A discrete convergence group action of a group
It was already observed by Gromov[6] that the natural action by translations of a word-hyperbolic group
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.
act as a discrete uniform convergence group on a compact metrizable space
is called geometric if this action is properly discontinuous and cocompact.
induces a uniform convergence action of
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.
is virtually a hyperbolic surface group, that is,
contains a finite index subgroup isomorphic to the fundamental group of a closed hyperbolic surface.
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