Convergence group

In mathematics , convergence groups are a term from the theory of dynamic systems that make it possible to investigate hyperbolic groups using dynamic (instead of geometric) methods.

definition

Let it be a group that acts continuously in a compact , metrizable space . The effect is called a convergence effect (and a convergence group) if the following condition is met: ${\displaystyle \Gamma }$ ${\displaystyle X}$ ${\displaystyle \Gamma }$

for every sequence there is a subsequence and two points , so that converges uniformly to on compact sets . ${\displaystyle (g_{n})_{n}\subset \Gamma }$ ${\displaystyle (g_{n_{i}})_{i}}$ ${\displaystyle a,b\in X}$ ${\displaystyle g_{n_{i}}\vert _{X\setminus \left\{a\right\}}}$ ${\displaystyle b}$

The last condition means: for every open environment of and every compact subset there is one with for all . ${\displaystyle U}$ ${\displaystyle b}$ ${\displaystyle K\subset X\setminus \left\{a\right\}}$ ${\displaystyle i_{0}}$ ${\displaystyle g_{n_{i}}(K)\subset U}$ ${\displaystyle i\geq i_{0}}$

An equivalent condition is that actually discontinuous on the space of the triplet ${\displaystyle \Gamma }$

${\displaystyle T(X)=\left\{(a,b,c)\in X^{3}:a\not =b,a\not =c,b\not =c\right\}}$

works.

Classification of elements

A nontrivial element of a convergence group acting on a compact, metric space is of exactly one of the following three types: ${\displaystyle \gamma }$ ${\displaystyle \Gamma }$

• elliptical : has finite order, ${\displaystyle \gamma }$
• parabolic : has infinite order and exactly one fixed point, ${\displaystyle \gamma }$
• loxodromic : has infinite order and exactly two fixed points. ${\displaystyle \gamma }$

For have and the same type. ${\displaystyle k\not =0}$ ${\displaystyle \gamma }$ ${\displaystyle \gamma ^{k}}$

If is parabolic with a fixed point , then applies to all . ${\displaystyle \gamma }$ ${\displaystyle a}$ ${\displaystyle \lim _{n\to \pm \infty }\gamma ^{n}x=a}$ ${\displaystyle x\in X}$

If is loxodromic with fixed points , then holds for all and for all and this convergence is uniform on compact subsets of . ${\displaystyle \gamma }$ ${\displaystyle a_{\pm }}$ ${\displaystyle \lim _{n\to \infty }\gamma ^{n}x=a_{-}}$ ${\displaystyle x\not =a_{+}}$ ${\displaystyle \lim _{n\to -\infty }\gamma ^{n}x=a_{+}}$ ${\displaystyle x\not =a_{-}}$ ${\displaystyle X\setminus \left\{a_{-},a_{+}\right\}}$

Limes amount

The limit set of is a minimal, non-empty, closed, -invariant subset . The convergence group is called non-elementary if it consists of more than two points. In this case a perfect set and especially infinite. ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle \Lambda (\Gamma )\subset X}$ ${\displaystyle \Lambda (\Gamma )}$ ${\displaystyle \Lambda (\Gamma )}$

The effect of convergence is called minimal if . ${\displaystyle \Lambda (\Gamma )=X}$

A conical boundary point is a point to which there is a sequence of different elements and points with and converges uniformly on compacta against the mapping, which is constant . For example, fixed points of a loxodromic image are conical boundary points. ${\displaystyle x\in X}$ ${\displaystyle \gamma _{n}\in \Gamma }$ ${\displaystyle a_{\pm }\in X}$ ${\displaystyle \lim _{n\to \infty }\gamma _{n}x=a_{+}}$ ${\displaystyle \gamma _{n}\mid _{X\setminus \left\{x\right\}}}$ ${\displaystyle a_{-}}$

Hyperbolic Groups, Uniform Convergence Groups

A convergence group is called an even convergence group or a uniform convergence group if the effect on is additionally co-compact . An equivalent condition is that every Limes point is a conical Limes point. ${\displaystyle TX}$

Theorem (Bowditch) : A group acting on a perfect , compact , metric space is a uniform convergence group if and only if a hyperbolic group and the convergence effect is conjugated by means of an - equivariant homeomorphism to the effect of on the Gromov boundary . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma }$ ${\displaystyle \partial _{\infty }\Gamma }$

Applications

The convergence property was originally introduced in the context of Kleinian groups by Gehring-Martin in order to axiomatize the properties of the effect of a Kleinian group on its Limes set .

The effect of convergence of a hyperbolic group on its boundary at infinity makes it possible to prove many algebraic statements about hyperbolic groups without using "hyperbolic" geometry, for example when proving the JSJ decomposition or the local connected properties of the boundary at infinity.

Convergence effects played an important role in the proof of the Seifert fiber space conjecture : this could be traced back to the fact that convergence groups acting on the circle must be virtual Fuchssch, i.e. H. contain a Fuchs group as a subgroup of finite index . The latter property ( proved by Casson- Jungreis and Gabai ) also enables an alternative proof for the Nielsen realization problem (originally proven by Kerckhoff ) .

literature

• Frederick W. Gehring , Gaven J. Martin: Discrete convergence groups. In: Carlos A. Berenstein (Ed.): Complex analysis. Proceedings of the Special Year held at the University of Maryland, College Park, 1985-86 (= Lecture Notes in Mathematics. Vol. 1275). Volume 1. Springer, Berlin et al. 1987, ISBN 3-540-18356-6 , pp. 158-167, doi : 10.1007 / BFb0078350 .
• David Gabai : Convergence groups are Fuchsian groups. In: Annals of Mathematics . Vol. 136, No. 3 (Nov., 1992), pp. 447-510, JSTOR 2946597 .
• Andrew Casson , Douglas Jungreis: Convergence groups and Seifert fibered 3-manifolds. Invent. Math. 118 (1994), no. 3, 441-456, doi : 10.1007 / BF01231540 .
• Pekka Tukia: Convergence groups and Gromov's metric hyperbolic spaces. In: New Zealand Journal of Mathematics. Vol. 23, No. 2, 1994, ISSN  1171-6096 , pp. 157-187, Link zu Digitisat .
• Eric M. Freden: Negatively curved groups have the convergence property I. In: Annales Academiæ Scientiarum Fennicæ. Series A. 1: Mathematica. Vol. 20, No. 2, 1995, ISSN  0066-1953 , pp. 333-348, digitized version (PDF; 137.81 kB) .
• Brian H. Bowditch : Convergence groups and configuration spaces. In: John Cossey, Charles F. Miller, Walter D. Neumann, Michael Shapiro (eds.): Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, 1996. de Gruyter, Berlin et al. 1999, ISBN 3-11-016366-7 , pp. 23-54, online .
• Pekka Tukia: Conical limit points and uniform convergence groups. In: Journal for pure and applied mathematics. Issue 501, 1998, pp. 71-98, doi : 10.1515 / crll.1998.081 .
• Brian H. Bowditch: A topological characterization of hyperbolic groups. In: Journal of the American Mathematical Society. Vol. 11, No. 3, 1998, pp. 643-667, doi : 10.1090 / S0894-0347-98-00264-1 .

Individual evidence

1. ↑ Ilya Kapovich, Nadia Benakli: Boundaries of hyperbolic groups. In: Sean Cleary, Robert Gilman, Alexei G. Myasnikov, Vladimir Shpilrain (Eds.): Combinatorial and geometric group theory. AMS Special Session Combinatorial Group Theory, Nov 4-5, 2000, New York, New York. AMS Special Session Computational Group Theory, April 28-29, 2001, Hoboken, New Jersey (= Contemporary Mathematics. Vol. 296). American Mathematical Society, Providence RI 2002, ISBN 0-8218-2822-3 , pp. 39-93, digitized version (PDF; 488 kB) .