In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Tits alternative, named after
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Early life ...
, is an important theorem about the structure of
finitely generated linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a ...
s.
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 geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of ''polynomial'' growth, as those groups which have nilpotent subgroups of finite index.
Statemen ...
. 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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
''H'' of ''G'' either ''H'' is virtually solvable or ''H'' contains a
nonabelian free
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
(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 group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mo ...
s;
*
Out(Fn)
In mathematics, Out(''Fn'') is the outer automorphism group of a free group on ''n'' generators. These groups are at universal stage in geometric group theory, as they act on the set of presentations with n generators of any finitely generated g ...
;
*Certain groups of
birational transformations of
algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s.
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
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
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) is a racket sport derived from tennis but distinguished by its playing surface being atop a stationary table, rather than the Tennis court, court on which players stand. Either individually or in teams of ...
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