In
mathematics, the Tits alternative, named for
Jacques Tits, is an important theorem about the structure of
finitely generated linear groups.
Statement
The theorem, proven by Tits,
is stated as follows.
Consequences
A linear group is not
amenable if and only if it contains a non-abelian free group (thus the
von Neumann conjecture, while not true in general, holds for linear groups).
The Tits alternative is an important ingredient in the proof of
Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result for linear groups (it reduces it to the case of solvable groups, which can be dealt with by elementary means).
Generalizations
In
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these group ...
, a group ''G'' is said to satisfy the Tits alternative if for every
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
''H'' of ''G'' either ''H'' is virtually solvable or ''H'' contains a
nonabelian free
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
(in some versions of the definition this condition is only required to be satisfied for all
finitely generated subgroups of ''G'').
Examples of groups satisfying the Tits alternative which are either not linear, or at least not known to be linear, are:
*
Hyperbolic groups
*
Mapping class groups;
*
Out(Fn);
*Certain groups of
birational transformations of
algebraic surfaces.
Examples of groups not satisfying the Tits alternative are:
*the
Grigorchuk group;
*
Thompson's group ''F''.
Proof
The proof of the original Tits alternative
[ is by looking at the Zariski closure of in . If it is solvable then the group is solvable. Otherwise one looks at the image of in the Levi component. If it is noncompact then a ]ping-pong
Table tennis, also known as ping-pong and whiff-whaff, is a sport in which two or four players hit a lightweight ball, also known as the ping-pong ball, back and forth across a table using small solid rackets. It takes place on a hard table div ...
argument finishes the proof. If it is compact then either all eigenvalues of elements in the image of are roots of unity and then the image is finite, or one can find an embedding of in which one can apply the ping-pong strategy.
Note that the proof of all generalisations above also rests on a ping-pong argument.
References
{{reflist, 30em
Infinite group theory
Geometric group theory
Theorems in group theory