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, and a linear group is an abstract group that is isomorphic to a matrix group over a field K – in other words, admitting a faithful, finite-dimensional representation over K.

Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behaviour (for example finitely generated infinite torsion groups).

Definition and basic examples

A group G is said to be linear if there exists a field K, an integer d and an injective homomorphism from G to the general linear group GLd (K) (a faithful linear representation of dimension d over K): 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:

  1. The group GLn(K) itself;
  2. The special linear group SLn(K) (the subgroup of matrices with determinant 1);
  3. The group of invertible upper (or lower) triangular matrices
  4. If gi is a collection of elements in GLn(K) 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. (Some authors require that the group be represented as a closed subgroup of the GLn(C).) Books that follow this approach include Hall (2015) and Rossman (2002).

Classes of linear groups

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 (for example the unit group of a quaternion algebra is a classical group). Note that the projective groups associated to these groups are also linear, though less obviously. For example, the group PSL2(R) is not a group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the adjoint representation), which can be used in the general case.

Many Lie groups are linear, but not all of them. The universal cover of SL2(R) 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 (for example lattices or thin groups) are also examples of interesting linear groups.

Finite groups

A finite group G of order 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 ring K[G] by left (or right) multiplication is linear and faithful. The finite groups of Lie type (classical groups over finite fields) are an important family of finite simple groups, as they take up most of the slots in the classification of finite simple groups.

Finitely generated matrix groups

While example 4 above is too general to define a distinctive class (it includes all linear groups), 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 (for instance the group generated by is free).
  • 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.
  • Braid groups (which are defined as a finitely presented group) have faithful linear representation on a finite-dimensional complex vector space where the generators act by explicit matrices.[1]

Examples from geometry

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(R) and this realises the fundamental group as a Fuchsian group. A generalisation of this construction is given by the notion of a (G,X)-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.[2]

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:

The Tits alternative states that a linear group either contains a non-abelian free group or else is virtually solvable (that is, contains a solvable group of finite index). This has many further consequences, for example:

Examples of non-linear groups

It is not hard to give infinitely generated examples of non-linear groups: for example the infinite abelian 2-group (Z/2Z)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.

Representation theory

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 (including those which are not faithful). These questions are the object of representation theory. Salient parts of the theory include:

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 (for example through Margulis' superrigidity theorem and other rigidity results).

Notes
  1. Stephen J. Bigelow (December 13, 2000), "Braid groups are linear" (PDF), Journal of the American Mathematical Society, 14 (2): 471–486
  2. Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry (2015). 3–manifolds groups. EMS Series of Lectures in Mathematics. European Math. Soc. Section 9.6.
  3. Wehrfritz 1973, p. 15.
  4. Wehfritz 1973, p. 57.
  5. Alperin, Roger C. (1987). "An Elementary Account Of Selberg's Lemma". L'Enseignement Mathématique. 33.CS1 maint: ref=harv (link)
  6. Bestvina, Mladen (2004). "Questions in Geometric Group Theory" (PDF). Question 1.15. Retrieved 17 August 2016.
  7. Formanek, E.; Procesi, C. (1992). "The automorphism group of a free group is not linear". J. Algebra. 149: 494–499. doi:10.1016/0021-8693(92)90029-l.CS1 maint: ref=harv (link)

References
  • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666.
  • Rossmann, Wulf (2002), Lie Groups: An Introduction through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford University Press, ISBN 9780198596837.
  • Suprnenko, D.A. (1976). Matrix groups. Translations of mathematical monographs. 45. American Mathematical Society. ISBN 0-8218-1595-4.
  • Wehrfritz, B.A.F. (1973). Infinite linear groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 76. Springer-Verlag.CS1 maint: ref=harv (link)
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