Cannon Thurston illustration

Limes set of a quasi-fox group

Cannon-Thurston maps are used in mathematics in the theory of small groups. They allow complicated Limes sets to be represented as continuous images of a circle.

Cannon-Thurston conjecture

The following theorem was conjectured by Cannon and Thurston and proved in this generality by Mahan Mj .

An area group THAT CONDITION freely and properly discontinuously without parabolic elements on the three-dimensional hyperbolic space . Then the equivariate inclusion ${\displaystyle \pi _{1}S}$ ${\displaystyle \mathbb {H} ^{3}}$

${\displaystyle i\colon {\widetilde {S}}\to \mathbb {H} ^{3}}$

the universal superposition continuously on the ideal edge to a continuous mapping ${\displaystyle {\widetilde {S}}=\mathbb {H} ^{2}}$ ${\displaystyle \partial _{\infty }{\widetilde {S}}=\partial _{\infty }\mathbb {H} ^{2}}$

${\displaystyle i\cup \partial _{\infty }i\colon \mathbb {H} ^{2}\cup \partial _{\infty }\mathbb {H} ^{2}\to \mathbb {H} ^{3}\cup \partial _{\infty }\mathbb {H} ^{3}}$

continue.

For is if and only if and endpoints are at infinity of the same sheet or the same ideal complementary polygon of the end lamination of . ${\displaystyle p\not =q\in \partial _{\infty }\mathbb {H} ^{2}}$ ${\displaystyle \partial _{\infty }i(p)=\partial _{\infty }i(q)}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle \pi _{1}S\backslash \mathbb {H} ^{3}}$

Applications

• If the limit set a finitely generated Klein group connected , then it is locally connected .

• Let be a geometrically finite Klein group. If it is for a different Klein group a group isomorphism is, the parabolic elements on parabolic elements maps, then there is a surjective, continuous mapping of the limit set by the limit set by , the fixed points of each element on the fixed points of the corresponding element in mapping. ${\displaystyle \Gamma }$ ${\displaystyle \Gamma ^{\prime }}$ ${\displaystyle \Gamma \to \Gamma ^{\prime }}$ ${\displaystyle \Gamma }$ ${\displaystyle \Gamma ^{\prime }}$ ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle \Gamma ^{\prime }}$

• Is a hyperbolic 3-manifold finite volume overlying the circle with fiber fray . Then the limit set of is an -invariant Peano curve . ${\displaystyle M}$ ${\displaystyle S}$ ${\displaystyle \pi _{1}S}$ ${\displaystyle \pi _{1}M}$

Generalizations

A hyperbolic group works freely and actually discontinuously on a Gromov hyperbolic space . One can ask whether the orbit maps ${\displaystyle \Gamma }$ ${\displaystyle X}$

${\displaystyle i\colon \Gamma \to X}$

on the Gromov boundary to a continuous map

${\displaystyle i\cup \partial _{\infty }i\colon \Gamma \cup \partial _{\infty }\Gamma \to X\cup \partial _{\infty }X}$

let continue. If such a continuous continuation exists, one denotes

${\displaystyle \partial _{\infty }i\colon \partial _{\infty }\Gamma \to \partial _{\infty }X}$

as a Cannon-Thurston figure .

There are numerous examples in which a Cannon-Thurston map does not exist, see Baker-Riley and Matsuda-Oguni.

However, a Cannon-Thurston map exists for

• the inclusion of a normal divisor in a hyperbolic group, ${\displaystyle i\colon \Gamma \to G}$

or for

• the inclusion of a corner space in a tree of Gromov hyperbolic spaces, in which all inclusions of edge spaces in corner spaces are quasi-isometric embeddings .

literature

• W. Abikoff. Two theorems on totally degenerate Kleinian groups. Amer. J. Math. 98, pp. 109-118, 1976.
• WJ Floyd. Group completions and limit sets of Kleinian groups. Invent. Math. 57, pp. 205-218, 1980.
• J. Cannon and WP Thurston. Group Invariant Peano Curves. Geom. Topol. 11, p. 1315-1355, 2007. (preprint from 1985)
• Y. Minsky. On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds. J. Amer. Math. Soc. 7, pp. 539-588, 1994.
• RC Alperin, W. Dicks, and J. Porti. The boundary of the Gieseking tree in hyperbolic three-space. Topology Appl. 93, pp. 219-259, 1999.
• CT McMullen. Local connectivity, Kleinian groups and geodesics on the blowup of the torus. Invent. Math. 146, pp. 35-91, 2001.
• Bra Bowditch. The Cannon-Thurston map for punctured surface groups. Math. Z. 255, pp. 35-76, 2007.
• O. Baker and T. Riley. Cannon-Thurston maps do not alway exist. Forum of Mathematics, Sigma, 1, e3, 2013.
• M. Mj. Cannon-Thurston Maps for Surface Groups. Ann. of Math. 179 (1), pp. 1-80, 2014.
• M. Mj. Ending Laminations and Cannon-Thurston Maps, with an appendix by S. Das and M. Mj. Geom. Funct. Anal. 24, pp. 297-321, 2014.
• Y. Matsuda and S. Oguni. On Cannon-Thurston maps for relatively hyperbolic groups. Journal of Group Theory 17 (1), pp. 41-47, 2014.
• M. Mj. Cannon-Thurston Maps for Kleinian Groups. Forum Math.Pi 5, e1, 49 pp., 2017.
• M. Mj. Cannon-Thurston maps , Proceedings of International Congress of Mathematicians 2018.