Convergence group

In mathematics, a convergence group or a discrete convergence group is a group ${\displaystyle \Gamma }$ acting by homeomorphisms on a compact metrizable space ${\displaystyle M}$ in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary ${\displaystyle \mathbb {S} ^{2}}$ of the hyperbolic 3-space ${\displaystyle \mathbb {H} ^{3}}$. 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 ${\displaystyle \Gamma }$ 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

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.

Examples

• 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]).

Classification of elements in convergence groups

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

Uniform convergence groups

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

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

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 ${\displaystyle M\to \partial G}$.

Convergence actions on the circle

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.

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 ${\displaystyle \mathbb {S} ^{2}}$,[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.

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]

References

1. F. W. Gehring and G. J. Martin, Discrete quasiconformal groups I, Proceedings of the London Mathematical Society 55 (1987), 331–358
2. B. H. Bowditch, Convergence groups and configuration spaces. Geometric group theory down under (Canberra, 1996), 23–54, de Gruyter, Berlin, 1999.
3. B. H. Bowditch, Treelike structures arising from continua and convergence groups. Memoirs of the American Mathematical Society 139 (1999), no. 662.
4. P. Tukia, Conical limit points and uniform convergence groups. Journal für die Reine und Angewandte Mathematik 501 (1998), 71–98
5. B. Bowditch, A topological characterisation of hyperbolic groups. Journal of the American Mathematical Society 11 (1998), no. 3, 643–667
6. Gromov, Mikhail (1987). "Hyperbolic groups". In Gersten, Steve M. (ed.). Essays in group theory. Mathematical Sciences Research Institute Publications. 8. New York: Springer. pp. 75–263. doi:10.1007/978-1-4613-9586-7_3. ISBN 0-387-96618-8. MR 0919829.CS1 maint: ref=harv (link)
7. P. Tukia, On quasiconformal groups. Journal d'Analyse Mathématique 46 (1986), 318–346.
8. D. Gabai, Convergence groups are Fuchsian groups. Annals of Mathematics 136 (1992), no. 3, 447–510.
9. A. Casson, D. Jungreis, Convergence groups and Seifert fibered 3-manifolds. Inventiones Mathematicae 118 (1994), no. 3, 441–456.
10. E. Freden, Negatively curved groups have the convergence property. I. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 20 (1995), no. 2, 333–348.
11. James W. Cannon, 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
12. A. Yaman, A topological characterisation of relatively hyperbolic groups. Journal für die Reine und Angewandte Mathematik 566 (2004), 41–89
13. V. Gerasimov, Expansive convergence groups are relatively hyperbolic, Geometric and Functional Analysis (GAFA) 19 (2009), no. 1, 137–169
14. W.Jeon, I. Kapovich, C. Leininger, K. Ohshika, Conical limit points and the Cannon-Thurston map. Conformal Geometry and Dynamics 20 (2016), 58–80