convergence group

Let $\backslash Gamma$ 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 $\backslash Gamma$ is called a convergence group or a discrete convergence group for this action) if for every infinite distinct sequence of elements $\backslash gamma\_n\; \backslash in\; \backslash Gamma$ there exist a subsequence $\backslash gamma\_\{n\_k\},\; k=1,2,\backslash dots$ and points $a,b\backslash in\; M$ such that the maps $\backslash gamma\_\{n\_k\}\backslash big|\_\{M\backslash setminus\backslash \{a\backslash \}\}$ converge uniformly on compact subsets to the constant map sending $M\backslash setminus\backslash \{a\backslash \}$ to $b$. Here converging uniformly on compact subsets means that for every open neighborhood $U$ of $b$ in $M$ and every compact $K\backslash subset\; M\backslash setminus\; \backslash \{a\backslash \}$ there exists an index $k\_0\backslash ge\; 1$ such that for every $k\backslash ge\; k\_0,$ $\backslash gamma\_\{n\_k\}(K)\backslash subseteq\; U$. Note that the "poles" $a,\; b\backslash in\; M$ associated with the subsequence $\backslash gamma\_\{n\_k\}$ are not required to be distinct.

The above definition of convergence group admits a useful equivalent reformulation in terms of the action of $\backslash Gamma$ on the "space of distinct triples" of $M$.

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

Then the following equivalence is known to hold:{{cite book

| last1=Bowditch | first1=B. H. | authorlink1=Brian Bowditch

| chapter=Convergence groups and configuration spaces

| title=Geometric group theory down under (Canberra, 1996)

| pages=23–54

| series=De Gruyter Proceedings in Mathematics

| publisher=de Gruyter, Berlin

| date=1999

| doi=10.1515/9783110806861.23| isbn=9783110806861 }}

Let $\backslash Gamma$ 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 $\backslash Gamma$ on $\backslash Theta(M)$ is properly discontinuous.

• The action of a Kleinian group $\backslash Gamma$ on $\backslash mathbb\; S^2=\backslash partial\; \backslash mathbb\; H^3$ by Möbius transformations is a convergence group action.
• The action of a word-hyperbolic group $G$ by translations on its ideal boundary $\backslash partial\; G$ is a convergence group action.
• The action of a relatively hyperbolic group $G$ by translations on its Bowditch boundary $\backslash partial\; G$ is a convergence group action.
• Let $X$ be a proper geodesic Gromov-hyperbolic metric space and let $\backslash Gamma$ be a group acting properly discontinuously by isometries on $X$. Then the corresponding boundary action of $\backslash Gamma$ on $\backslash partial\; X$ is a discrete convergence action (Lemma 2.11 of ).

Let $\backslash Gamma$ be a group acting by homeomorphisms on a compact metrizable space $M$with at least three points, and let $\backslash gamma\backslash in\backslash Gamma$. Then it is known (Lemma 3.1 in or Lemma 6.2 in {{cite journal

| last1=Bowditch | first1=B. H. | authorlink1=Brian Bowditch

| title=Treelike structures arising from continua and convergence groups

| journal=Memoirs of the American Mathematical Society

| volume=139

| date=1999

| issue=662

| doi=10.1090/memo/0662}}) that exactly one of the following occurs:

(1) The element $\backslash gamma$ has finite order in $\backslash Gamma$; in this case $\backslash gamma$ is called elliptic.

(2) The element $\backslash gamma$ has infinite order in $\backslash Gamma$ and the fixed set $\backslash operatorname\{Fix\}\_M(\backslash gamma)$ is a single point; in this case $\backslash gamma$ is called parabolic.

(3) The element $\backslash gamma$ has infinite order in $\backslash Gamma$ and the fixed set $\backslash operatorname\{Fix\}\_M(\backslash gamma)$ consists of two distinct points; in this case $\backslash gamma$ is called loxodromic.

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

If $\backslash gamma\backslash in\; \backslash Gamma$ is parabolic with a fixed point $a\backslash in\; M$ then for every $x\backslash in\; M$ one has $\backslash lim\_\{n\backslash to\backslash infty\}\backslash gamma^nx=\backslash lim\_\{n\backslash to-\backslash infty\}\backslash gamma^nx\; =a$

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

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

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

An important result of Tukia,{{cite journal

| last1=Tukia | first1=Pekka

| title=Conical limit points and uniform convergence groups

| journal=Journal für die reine und angewandte Mathematik

| volume=1998

| date=1998

| issue=501

| pages=71–98

| doi=10.1515/crll.1998.081}} also independently obtained by Bowditch,{{cite journal

| last1=Bowditch | first1=Brian H.

| title=A topological characterisation of hyperbolic groups

| journal=Journal of the American Mathematical Society

| volume=11

| date=1998

| issue=3

| pages=643–667

| doi=10.1090/S0894-0347-98-00264-1 | doi-access=free}} states:

A discrete convergence group action of a group $\backslash Gamma$ on a compact metrizable space $M$ is uniform if and only if every non-isolated point of $M$ is a conical limit point.

It was already observed by Gromov{{cite book

| author-link=Mikhail Gromov (mathematician) | first=Mikhail | last=Gromov

| chapter=Hyperbolic groups

| title=Essays in group theory

| pages=75–263

| series=Mathematical Sciences Research Institute Publications

| volume=8

| publisher=Springer

| location=New York

| year=1987

| doi=10.1007/978-1-4613-9586-7_3 | doi-access=free

| mr=919829

| editor-last=Gersten

| editor-first=Steve M.

| isbn=0-387-96618-8 }} that the natural action by translations of a word-hyperbolic group $G$ on its boundary $\backslash partial\; G$ 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 $M\backslash to\; \backslash partial\; G$.

An isometric action of a group $G$ on the hyperbolic plane $\backslash mathbb\; H^2$ is called geometric if this action is properly discontinuous and cocompact. Every geometric action of $G$ on $\backslash mathbb\; H^2$ induces a uniform convergence action of $G$ on $\backslash mathbb\; S^1\; =\backslash partial\; H^2\backslash approx\; \backslash partial\; G$.

An important result of Tukia (1986),{{cite journal

| last1=Tukia | first1=Pekka | authorlink1=Pekka Tukia

| title=On quasiconformal groups

| journal=Journal d'Analyse Mathématique

| volume=46

| date=1986

| pages=318–346

| doi=10.1007/BF02796595 | doi-access=}} Gabai (1992),{{cite journal

| last1=Gabai | first1=Davis | authorlink1=David Gabai

| title=Convergence groups are Fuchsian groups

| journal=Annals of Mathematics

| series=Second series

| volume=136

| date=1992

| issue=3

| pages=447–510

| doi=10.2307/2946597

| jstor=2946597| url=http://projecteuclid.org/euclid.bams/1183657188 }} Casson–Jungreis (1994),{{cite journal

| last1=Casson | first1=Andrew

| last2=Jungreis | first2=Douglas

| title=Convergence groups and Seifert fibered 3-manifolds

| journal=Inventiones Mathematicae

| volume=118

| date=1994

| issue=3

| pages=441–456

| doi=10.1007/BF01231540 | bibcode=1994InMat.118..441C

| doi-access=}} and Freden (1995){{cite journal

| last1=Freden | first1=Eric M.

| title=Negatively curved groups have the convergence property I

| journal=Annales Academiae Scientiarum Fennicae

| series=Series A

| volume=20

| date=1995

| issue=2

| pages=333–348

| url=https://www.acadsci.fi/mathematica/Vol20/freden.pdf

| access-date=September 12, 2022}} shows that the converse also holds:

Theorem. If $G$ is a group acting as a discrete uniform convergence group on $\backslash mathbb\; S^1$ then this action is topologically conjugate to an action induced by a geometric action of $G$ on $\backslash mathbb\; H^2$ by isometries.

Note that whenever $G$ acts geometrically on $\backslash mathbb\; H^2$, 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.

One of the equivalent reformulations of Cannon's conjecture, (posed by James W. Cannon,{{cite book

| last1=Cannon | first1=James W.

| chapter=The theory of negatively curved spaces and groups

| title=Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989)

| pages=315–369

| publisher=Oxford Sci. Publ., Oxford Univ. Press, New York

| date=1991

| chapter-url=http://mmontee.people.sites.carleton.edu/Cannon_Negatively_Curved.pdf

| access-date=September 12, 2022}}

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 {{Citation

last1=Martin | first1=Gaven J.

| last2=Skora | first2=Richard K.

| title=Group Actions of the 2-Sphere

| journal=American Journal of Mathematics

| volume=111

| pages=387–402

| publisher=The Johns Hopkins University Press

| date=1989

}})

These conjectures are in terms of word-hyperbolic groups with boundaries homeomorphic to $\backslash mathbb\; S^2$, says that if $G$ is a group acting as a discrete uniform convergence group on $\backslash mathbb\; S^2$ then this action is topologically conjugate to an action induced by a geometric action of $G$ on $\backslash mathbb\; H^3$ by isometries. These conjectures still remains open.

• Yaman gave a characterization of relatively hyperbolic groups in terms of convergence actions,{{cite journal

| last1=Yaman | first1=Asli

| title=A topological characterisation of relatively hyperbolic groups

| journal=Journal für die reine und angewandte Mathematik

| volume=2004

| date=2004

| issue=566

| pages=41–89

| doi=10.1515/crll.2004.007}} 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.{{cite journal

| last1=Gerasimov | first1=Victor

| title=Expansive convergence groups are relatively hyperbolic

| journal=Geometric and Functional Analysis

| volume=19

| date=2009

| issue=1

| pages=137–169

| doi=10.1007/s00039-009-0718-7 | doi-access=}}

• 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.{{cite journal

| last1=Jeon | first1=Woojin

| last2=Kapovich | first2=Ilya | authorlink2=Ilya Kapovich

| last3=Leininger | first3=Christopher

| last4=Ohshika | first4=Ken'ichi

| title=Conical limit points and the Cannon-Thurston map

| journal=Conformal Geometry and Dynamics

| volume=20

| date=2016

| issue=4

| pages=58–80

| arxiv=1401.2638

| doi=10.1090/ecgd/294 | doi-access=free}}

{{Reflist}}

Category:Group theory

Category:Dynamical systems

Category:Geometric topology

Category:Geometric group theory