A group G is said to be linear if there exists a field K, an integer d and an injectivemorphism from G to the general linear group GLd : if needed one can mention the field and dimension by saying that G is linear of degree d over K. Basic instances are groups which are defined as subgroups of a linear group, for example:
If gi is a collection of elements in GLn indexed by a set I, then the subgroup generated by the gi is a linear group.
In the study of Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of complex numbers. Books that follow this approach include Hall and Rossman.
Classes of linear groups
Classical groups and related examples
The so-called classical groups generalise the examples 1 and 2 above. They arise as linear algebraic groups, that is, as subgroups of GLn defined by a finite number of equations. Basic examples are orthogonal, unitary and symplectic groups but it is possible to construct more using division algebras. Note that the projective groups associated to these groups are also linear, though less obviously. For example, the group PSL2 is not a group of 2×2 matrices, but it has a faithful representation as 3×3 matrices, which can be used in the general case. Many Lie groups are linear but not all of them. The universal cover of SL2 is not linear, as are many solvable groups, for instance the quotient of the Heisenberg group by a central cyclic subgroup. Discrete subgroups of classical Lie groups are also examples of interesting linear groups.
Finite groups
A finite group G of cardinality n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact that the action of G on the group ringK by left multiplication is linear and faithful. The finite groups of Lie type are an important family of finite simple group, as they take up most of the slots in the classification of finite simple groups.
While example 4 above is too general to define a distinctive class, restricting to a finite index set I, that is, to finitely generated groups allows to construct many interesting examples. For example:
The ping-pong lemma can be used to construct many examples of linear groups which are free groups.
Arithmetic groups are known to be finitely generated. On the other hand, it is a difficult problem to find an explicit set of generators for a given arithmetic group.
In some cases the fundamental group of a manifold can be shown to be linear by using representations coming from a geometric structure. For example, all closed surfaces of genus at least 2 are hyperbolic Riemann surfaces. Via the uniformisation theorem this gives rise to a representation of its fundamental group in the isometry group of the hyperbolic plane, which is isomorphic to PSL2 and this realises the fundamental group as a Fuchsian group. A generalisation of this construction is given by the notion of a -structure on a manifold. Another example is the fundamental group of Seifert manifolds. On the other hand, it is not known whether all fundamental groups of 3-manifolds are linear.
Properties
While linear groups are a vast class of examples, among all infinite groups they are distinguished by many remarkable properties. Finitely generated linear groups have the following properties:
It is not hard to give infinitely generated examples of non-linear groups: for example the infinite abelian 2-group N cannot be linear since if this was the case it would be diagonalisable and finite. Since the symmetric group on an infinite set contains this group it is also not linear. Finding finitely generated examples is subtler and usually requires the use of one of the properties listed above.
Since any finitely linear group is residually finite, it cannot be both simple and infinite. Thus finitely generated infinite simple groups, for example Thompson's groupF, and Higman's group, are not linear.
By the corollary to the Tits alternative mentioned above, groups of intermediate growth such as Grigorchuk's group are not linear.
By Burnside's theorem an infinite, finitely generated torsion groups such as Tarski monster groups cannot be linear.
There are examples of hyperbolic groups which are not linear, obtained as quotients of lattices in the Lie groups Sp.
Once a group has been established to be linear it is interesting to try to find "optimal" faithful linear representations for it, for example of the lowest possible dimension, or even to try and classify all its linear representations. These questions are the object of representation theory. Salient parts of the theory include:
Representation theory of finite groups;
Representation theory of Lie groups and more generally linear algebraic groups.
The representation theory of infinite finitely generated groups is in general mysterious; the object of interest in this case are the character varieties of the group, which are well understood only in very few cases, for example free groups, surface groups and more generally lattices in Lie groups.
