Tits alternative

In the mathematical fields of group theory and linear algebra , the Tits alternative describes a property of matrix groups , namely either to be almost resolvable or to contain a non-Abelian free subgroup. It is named after the Belgian-French mathematician Jacques Tits .

Tits alternative

Let it be an arbitrary field and the general linear group , i. H. the group of invertible matrices with entries from the body . ${\displaystyle K}$ ${\displaystyle GL(n,K)}$ ${\displaystyle K}$

Then exactly one of the following two alternatives applies to each finitely generated subgroup : ${\displaystyle \Gamma \subset GL(n,K)}$

• ${\displaystyle \Gamma }$is almost dissolvable , d. H. it contains a resolvable subgroup of finite index , or
• ${\displaystyle \Gamma }$contains a free subgroup of rank . ${\displaystyle F}$ ${\displaystyle rk(F)\geq 2}$

The two possibilities are mutually exclusive.

This theorem was conjectured by Bass and Serre and proved by Jacques Tits in 1972 . An essential ingredient in the proof was the ping-pong lemma .

In general, it is said that a class of groups fulfills the Tits alternative if all groups from this class are either almost resolvable or contain a free subgroup of rank . ${\displaystyle \geq 2}$

Examples

The Tits alternative applies to numerous classes of groups, including the following:

• Finally created subgroups of for any body ${\displaystyle GL(n,K)}$ ${\displaystyle K}$
• Hyperbolic groups
• Figure class groups of compact areas
• ${\displaystyle Out(F_{n})}$
• Groups which are free and actually on a CAT (0) - Dice complex act
• Groups of polynomial automorphisms of the ${\displaystyle \mathbb {C} ^{2}}$
• Groups of bimeromorphic automorphisms of compact Kahler manifolds
• Groups of birational images of compact Kähler areas

Counterexamples

A group to which the Tits alternative does not apply must either

• an indirect group , but not almost-dissolvable

or but

• not be indirect, but not contain a free subgroup of rank . ${\displaystyle \geq 2}$

Groups with either of these characteristics are difficult to construct and are considered exotic. A number of examples are now known:

• the Thompson group
• the Tarski group
• the Burnside groups for odd ${\displaystyle B(m,n)}$ ${\displaystyle n\geq 665}$
• the Grigorchuk group .

There are finitely generated resolvable groups whose automorphism groups do not satisfy the Tits alternative.

literature

Tits, J .: "Free subgroups in linear groups". Journal of Algebra 20 (2), 250-270 (1972). online (pdf)

Web links

Tointon: The Tits Alternative

Individual evidence

1. ↑ Gromov, M .: Hyperbolic groups. Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987).
2. ^ Ivanov, NV: Algebraic properties of the Teichmüller modular group. So V. Math., Dokl. 29: 288-291 (1984); McCarthy, John: A "Tits-alternative" for subgroups of surface mapping class groups. Trans. Am. Math. Soc. 291: 583-612 (1985).
3. ↑ Bestvina, Mladen; Feighn, Mark; Handel, Michael: The Tits alternative for Out (F _n ). I: Dynamics of exponentially-growing automorphisms. Ann. Math. (2) 151, No. 2: 517-623 (2000). The Tits alternative for Out (F _n ). II: A Kolchin type theorem. Ann. Math. (2) 161, No. 1, 1-59 (2005).
4. ↑ Sageev, Michah; Wise, Daniel T .: The Tits alternative for CAT (0) cubical complexes. Bull. Lond. Math. Soc. 37, no. 5, 706-710 (2005).
5. ↑ Lamy, Stéphane: The Tits alternative for Aut [ℂ2]. (L'alternative de Tits pour Aut [ℂ ² ].) J. Algebra 239, no. 2: 413-437 (2001).
6. ↑ Oguiso, Keiji: Tits alternative in hyper-Kähler manifolds. Math. Res. Lett. 13, No. 2-3, 307-316 (2006).
7. ^ Cantat, Serge: Sur les groupes de transformations birationnelles des surfaces. Ann. Math. (2) 174, No. 1, 299-340 (2011).
8. ^ Hartley, Brian: A conjecture of Bachmuth and Mochizuki on automorphisms of soluble groups. Can. J. Math. 28: 1302-1310 (1976).