Convergence group explained

\Gamma acting by homeomorphisms on a compact metrizable space M in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary S ^{2} of the hyperbolic 3-space 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

\Gamma be a group acting by homeomorphisms on a compact metrizable space M . This action is called a convergence action or a discrete convergence action (and then \Gamma is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements \gamma _{n} \in \Gamma there exist a subsequence \gamma _{n _{k}} , k =1 , 2 , ... and points a,b\in M such that the maps \gamma _{n _{k}} | _{M\setminus\{a\} } converge uniformly on compact subsets to the constant map sending M\setminus\{a\} to b . Here converging uniformly on compact subsets means that for every open neighborhood U of b in M and every compact K\subset M\setminus \{a\} there exists an index k _{0\ge} 1 such that for every k\ge k _{0,} \gamma _{n _{k}} (K)\subseteq U . Note that the "poles" a, b\in M associated with the subsequence \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

\Gamma on the "space of distinct triples" of M .For a set M denote \Theta(M): = M ^{3\setminus} \Delta(M) , where \Delta(M) = \{(a,b,c)\in M ^{3\mid} \#\{a,b,c\}\le 2 \} . The set \Theta(M) is called the "space of distinct triples" for M .

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

Let

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

Examples

\Gamma on S ^{2=\partial} H ^{3} by Möbius transformations is a convergence group action.G by translations on its ideal boundary \partial G is a convergence group action.G by translations on its Bowditch boundary \partial G is a convergence group action.

• Let

X be a proper geodesic Gromov-hyperbolic metric space and let \Gamma be a group acting properly discontinuously by isometries on X . Then the corresponding boundary action of \Gamma on \partial X is a discrete convergence action (Lemma 2.11 of).

Classification of elements in convergence groups

Let

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

(1) The element

\gamma has finite order in \Gamma ; in this case \gamma is called elliptic.

(2) The element

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

(3) The element

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

Moreover, for every

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

If

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

Uniform convergence groups

A discrete convergence action of a group

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

Conical limit points

Let

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

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

A discrete convergence group action of a group

\Gamma on a compact metrizable space M is uniform if and only if every non-isolated point of M is a conical limit point.

Word-hyperbolic groups and their boundaries

G on its boundary \partial G 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

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

Convergence actions on the circle

An isometric action of a group

G on the hyperbolic plane H ^{2} is called geometric if this action is properly discontinuous and cocompact. Every geometric action of G on H ^{2} induces a uniform convergence action of G on S ^{1} = \partial H ^{2 ≈} \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

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

Note that whenever

G acts geometrically on H ^{2} , the group G is virtually a hyperbolic surface group, that is, 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, originally posed by James W. Cannon in terms of word-hyperbolic groups with boundaries homeomorphic to

S ^{2} ,^[11] says that if G is a group acting as a discrete uniform convergence group on S ^{2} then this action is topologically conjugate to an action induced by a geometric action of G on H ^{3} by isometries. This conjecture still remains open.

Applications and further generalizations

• 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]

Notes and References

1. Gehring . F. W.. Martin . G. J.. Discrete quasiconformal groups I. Proceedings of the London Mathematical Society. 55. 1987. 2. 331–358. 10.1093/plms/s3-55_2.331. 2027.42/135296. free.
2. Book: Bowditch . B. H. . Brian Bowditch. Convergence groups and configuration spaces. Geometric group theory down under (Canberra, 1996). 23–54. De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. 1999. 10.1515/9783110806861.23. 9783110806861 .
3. Bowditch . B. H. . Brian Bowditch. Treelike structures arising from continua and convergence groups. Memoirs of the American Mathematical Society. 139. 1999. 662. 10.1090/memo/0662.
4. Tukia . Pekka. Conical limit points and uniform convergence groups. Journal für die reine und angewandte Mathematik. 1998. 1998. 501. 71–98. 10.1515/crll.1998.081.
5. Bowditch . Brian H.. A topological characterisation of hyperbolic groups. Journal of the American Mathematical Society. 11. 1998. 3. 643–667. 10.1090/S0894-0347-98-00264-1 . free.
6. Book: Gromov , Mikhail . Mikhail Gromov (mathematician) . Hyperbolic groups. Essays in group theory. 75–263. Mathematical Sciences Research Institute Publications. 8. Springer. New York. 1987. 10.1007/978-1-4613-9586-7_3 . free. 919829. Gersten. Steve M.. 0-387-96618-8 .
7. Tukia . Pekka . Pekka Tukia. On quasiconformal groups. Journal d'Analyse Mathématique. 46. 1986. 318–346. 10.1007/BF02796595 .
8. Gabai . Davis . David Gabai. Convergence groups are Fuchsian groups. Annals of Mathematics. Second series. 136. 1992. 3. 447–510. 10.2307/2946597. 2946597.
9. Casson . Andrew. Jungreis . Douglas. Convergence groups and Seifert fibered 3-manifolds. Inventiones Mathematicae. 118. 1994. 3. 441–456. 10.1007/BF01231540 . 1994InMat.118..441C.
10. Freden . Eric M.. Negatively curved groups have the convergence property I. Annales Academiae Scientiarum Fennicae. Series A. 20. 1995. 2. 333–348. September 12, 2022.
11. Book: Cannon . James W.. The theory of negatively curved spaces and groups. Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989). 315–369. Oxford Sci. Publ., Oxford Univ. Press, New York. 1991. http://mmontee.people.sites.carleton.edu/Cannon_Negatively_Curved.pdf. September 12, 2022.
12. Yaman . Asli. A topological characterisation of relatively hyperbolic groups. Journal für die reine und angewandte Mathematik. 2004. 2004. 566. 41–89. 10.1515/crll.2004.007.
13. Gerasimov . Victor. Expansive convergence groups are relatively hyperbolic. Geometric and Functional Analysis. 19. 2009. 1. 137–169. 10.1007/s00039-009-0718-7 .
14. Jeon . Woojin. Kapovich . Ilya . Ilya Kapovich. Leininger . Christopher. Ohshika . Ken'ichi. Conical limit points and the Cannon-Thurston map. Conformal Geometry and Dynamics. 20. 2016. 4. 58–80. 1401.2638. 10.1090/ecgd/294 . free.