In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Tits alternative, named for
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.
Life and ...
, 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 faithf ...
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 In mathematics, the von Neumann conjecture stated that a group (mathematics), group ''G'' is non-Amenable group, amenable if and only if ''G'' contains a subgroup that is a free group on two Generating set of a group, generators. The conjecture was ...
, 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.
Statement ...
. 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 group (mathematics), groups and topology, topological and geometry, geometric pro ...
, 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 subgroup ...
''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 subgroup ...
(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 group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s
*
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.
Mot ...
s;
*
Out(Fn)
In mathematics, Out(''Fn'') is the outer automorphism group of a free group on ''n'' generators. These groups play an important role in geometric group theory.
Outer space
Out(''Fn'') acts geometrically on a cell complex known as Culler–Vog ...
;
*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 In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, f ...
;
*
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 which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is no ...
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