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Linear group
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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 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 behavior (for example, finitely generated infinite torsion groups).
Definition and basic examples
[edit]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:
- The group GLn(K) itself;
- The special linear group SLn(K) (the subgroup of matrices with determinant 1);
- The group of invertible upper (or lower) triangular matrices
- 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)[1] and Rossmann (2002).[2]
Classes of linear groups
[edit]Classical groups and related examples
[edit]The so-called classical groups generalize 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
[edit]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
[edit]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.[3] The mapping class group of a genus 2 surface is also known to be linear.[4]
Examples from geometry
[edit]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 uniformization 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 realizes the fundamental group as a Fuchsian group. A generalization 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.[5]
Properties
[edit]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:
- They are residually finite;
- Burnside's theorem: a torsion group of finite exponent which is linear over a field of characteristic 0 must be finite;[6]
- Schur's theorem: a torsion linear group is locally finite. In particular, if it is finitely generated then it is finite.[7]
- Selberg's lemma: any finitely generated linear group contains a torsion-free subgroup of finite index.[8]
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:
- the Dehn function of a finitely generated linear group can only be either polynomial or exponential;
- an amenable linear group is virtually solvable, in particular elementary amenable;
- the von Neumann conjecture is true for linear groups.
Examples of non-linear groups
[edit]It is not hard to give infinitely generated examples of non-linear groups: for example the infinite abelian group (Z/2Z)N x (Z/3Z)N cannot be linear.[9] 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 group F, and the quotient of Higman's group by a maximal proper normal subgroup, are not linear.
- By the corollary to the Tits alternative mentioned above, groups of intermediate growth such as Grigorchuk's group are not linear.
- Again by the Tits alternative, as mentioned above all counterexamples to the von Neumann conjecture are not linear. This includes Thompson's group F and Tarski monster groups.
- By Burnside's theorem, 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(n, 1).[10]
- The outer automorphism group Out(Fn) of the free group is known not to be linear for n at least 4.[11]
- In contrast with the case of braid groups, it is an open question whether the mapping class group of a surface of genus > 2 is linear.
Representation theory
[edit]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 to 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:
- 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 (for example through Margulis' superrigidity theorem and other rigidity results).
Notes
[edit]- ^ Hall (2015)
- ^ Rossmann (2002)
- ^ Stephen J. Bigelow (December 13, 2000), "Braid groups are linear" (PDF), Journal of the American Mathematical Society, 14 (2): 471–486, doi:10.1090/S0894-0347-00-00361-1, S2CID 18936096
- ^ Bigelow, Stephen J.; Budney, Ryan D. (2001), "The mapping class group of a genus two surface is linear", Algebraic and Geometric Topology, 1 (2): 699–708, arXiv:math/0010310, doi:10.2140/agt.2001.1.699
- ^ Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry (2015). 3–manifolds groups. EMS Series of Lectures in Mathematics. European Math. Soc. Section 9.6.
- ^ Wehrfritz 1973, p. 15.
- ^ Wehrfritz 1973, p. 57.
- ^ Alperin, Roger C. (1987). "An Elementary Account Of Selberg's Lemma". L'Enseignement Mathématique. 33.
- ^ This follows from Wehrfritz (1973, Theorem 2.2).
- ^ Bestvina, Mladen (2004). "Questions in Geometric Group Theory" (PDF). Question 1.15. Retrieved 17 August 2016.
- ^ Formanek, E.; Procesi, C. (1992). "The automorphism group of a free group is not linear". J. Algebra. 149 (2): 494–499. doi:10.1016/0021-8693(92)90029-l.
References
[edit]- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 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. Vol. 45. American Mathematical Society. ISBN 0-8218-1595-4.
- Wehrfritz, B.A.F. (1973). Infinite linear groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 76. Springer-Verlag.
Linear group
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In mathematics, a linear group is a group that admits a faithful finite-dimensional representation over a field , meaning it is isomorphic to a subgroup of the general linear group for some positive integer and field .[1] The general linear group consists of all invertible matrices with entries in , under the operation of matrix multiplication, with the identity matrix serving as the group identity.[2] This structure allows linear groups to model transformations of vector spaces while preserving the group axioms of associativity, identity, and inverses.[3]
Prominent examples of linear groups include the special linear group , comprising matrices in with determinant 1, which preserves volume in the associated vector space.[2] Over the real numbers , the orthogonal group consists of matrices satisfying , preserving the Euclidean inner product, while its intersection with yields the special orthogonal group .[1] For complex numbers , the unitary group includes matrices with (where is the conjugate transpose), and combines unitarity with determinant 1.[1]
Linear groups are central to representation theory, where every finite-dimensional representation of a group yields a linear group homomorphism into , facilitating the study of group actions on vector spaces.[1] In the context of Lie groups, closed linear groups (topologically closed subgroups of or ) possess associated Lie algebras, consisting of matrices such that for all real , enabling differential analysis of their structure.[1] Over algebraically closed fields, linear algebraic groups—affine varieties equipped with group operations—are precisely the closed subgroups of some , playing a key role in algebraic geometry and the classification of semisimple groups.[3]
Definition and Fundamentals
Formal Definition
In linear algebra and group theory, the general linear group is defined as the set of all invertible matrices with entries from a field , where the group operation is matrix multiplication. This group captures the structure of all invertible linear transformations of a -dimensional vector space over .[2] A group is said to be linear if there exists a field , a positive integer , and an injective group homomorphism , making isomorphic to its image, a subgroup of . Equivalently, admits a faithful finite-dimensional representation over , where the representation is a homomorphism into the group of linear automorphisms of a finite-dimensional vector space, realized concretely as matrix multiplication. This formulation emphasizes that linear groups are precisely those abstract groups that can be embedded as matrix groups acting linearly on vector spaces.[4] The concept of linear groups originated in the work of Camille Jordan, who introduced the term in his 1870 treatise Traité des substitutions et des équations algébriques to describe groups arising from linear substitutions in the context of solving algebraic equations via Galois theory. Jordan's approach connected permutation groups to linear representations, laying foundational groundwork for modern representation theory.[5]Basic Properties
A fundamental property of linear groups is that every finite group is linear. By Cayley's theorem, any finite group of order embeds as a subgroup of the symmetric group , and in turn embeds into via permutation matrices, whose entries are rational integers, establishing that is isomorphic to a subgroup of .[6] For a group to be linear over a field , there exists a faithful representation for some , where the representation is faithful if its kernel is trivial, i.e., .[7] Over algebraically closed fields such as , linear groups admit faithful complex representations, while over , they admit faithful real representations.[8] The degree of linearity of a group is the minimal integer such that embeds into for some field . For finite groups, this degree satisfies , as the permutation representation provides an upper bound.[9]Classifications and Examples
Classical Groups
The classical groups form a foundational family of linear algebraic groups defined over a field , consisting of subgroups of the general linear group that preserve specific non-degenerate bilinear forms on the underlying vector space. These groups arise naturally in the study of linear transformations that maintain geometric structures such as inner products or symplectic pairings, and they play a central role in representation theory and Lie theory. The general linear group itself, comprising all invertible matrices over , serves as the ambient space, while its determinant-1 subgroup, the special linear group , preserves oriented volume and provides a key example of a classical group with dimension .[10] Orthogonal groups preserve symmetric bilinear forms, specifically quadratic forms. The orthogonal group consists of matrices satisfying , where is the identity matrix, thus preserving the standard quadratic form (or its associated bilinear form ). The special orthogonal group is the kernel of the determinant map on , yielding transformations of determinant 1 and dimension . A prominent real example is , which includes all rotations and reflections in three-dimensional Euclidean space, with specifically realizing the group of proper rotations.[11][10] Symplectic groups act on even-dimensional spaces while preserving skew-symmetric bilinear forms. For dimension , the symplectic group comprises matrices such that , where defines the standard symplectic form . This group has dimension and captures transformations that maintain the symplectic structure essential in Hamiltonian mechanics and quantum theory.[11][10] Unitary groups, defined over the complex numbers , preserve Hermitian forms. The unitary group consists of matrices satisfying , where is the conjugate transpose, thus preserving the form with . The special unitary group restricts to determinant 1; U(n, ℂ) has dimension n², while SU(n, ℂ) has dimension n² - 1, both being compact Lie groups. These real compact forms, such as and , embed into the broader framework of real Lie groups, where their Lie algebras (e.g., skew-symmetric matrices for ) govern infinitesimal transformations via the exponential map.[11][10]Finite Groups
Finite subgroups of the general linear group , where is a field, are known as finite linear groups. These groups arise naturally in the study of symmetries that preserve linear structures, and their representation theory provides deep insights into their structure. Over fields of characteristic zero, such as or , every finite group admits a faithful linear representation, embedding it as a subgroup of for some . Specifically, by realizing the group via its action on itself, the degree of such a representation is at most the order of the group, .[12][13] A concrete realization of this linearity is the permutation representation, where the group acts on a vector space with basis corresponding to its elements. For the symmetric group , this embeds into via permutation matrices, which are integer matrices with exactly one 1 in each row and column, preserving the rational field. The alternating group , as a subgroup of index 2 in , inherits a similar embedding into , acting on the same space but with even permutations. These examples illustrate how classical permutation groups become linear over .[13][12] Over the complex numbers , the representation theory of finite linear groups simplifies significantly due to complete reducibility. Maschke's theorem states that if is a finite group and is a field whose characteristic does not divide , then every finite-dimensional representation of over is completely reducible, meaning it decomposes as a direct sum of irreducible representations. This semisimplicity allows any representation to be analyzed through its irreducible constituents.[13][12] Character theory provides a powerful tool for studying these irreducible representations. The character of a representation is the class function defined by , which is constant on conjugacy classes and determines the representation up to isomorphism. For irreducible representations over , the irreducible characters form an orthonormal basis for the space of class functions under the inner product . The number of irreducible representations equals the number of conjugacy classes, and their degrees satisfy the orthogonality relations, enabling computations of decomposition multiplicities. A key property is that the dimension of any irreducible representation divides the group order: This divisibility follows from the structure of the group algebra and Frobenius reciprocity.[13][12]Finitely Generated and Infinite Groups
Finitely generated linear groups that are infinite provide a rich class of examples where the algebraic structure interacts profoundly with geometric and analytic properties. These groups arise as discrete subgroups of GL(n, K) for some field K, often with K = ℚ or ℝ, and their finite generation implies a countable presentation that contrasts with the continuity of the ambient Lie group. Unlike finite linear groups, infinite ones exhibit unbounded word lengths and can display hyperbolic dynamics or arithmetic rigidity, depending on the context.[14] A prominent example involves free groups, which embed into SL(2, ℤ) through Schottky subgroups, leveraging the ping-pong lemma to ensure freeness. Schottky groups in SL(2, ℤ) are generated by hyperbolic elements whose fundamental domains on the hyperbolic plane satisfy disjoint translation conditions, yielding a free action on the boundary circle that produces free subgroups of arbitrary rank. This embedding, established via the ping-pong mechanism, demonstrates that non-abelian free groups act faithfully and discretely within low-dimensional special linear groups over the integers.[15][16] Arithmetic groups offer another canonical family of finitely generated infinite linear groups, exemplified by SL(n, ℤ) as a lattice in SL(n, ℝ). These groups consist of integer matrices preserving volume and form discrete subgroups of finite covolume in the real Lie group, with rich modular representations arising from their action on symmetric spaces. Congruence subgroups, such as the principal congruence subgroup Γ(N) = {g ∈ SL(n, ℤ) | g ≡ I mod N}, provide finite-index normal subgroups that are also arithmetic and finitely generated, facilitating the study of modular forms and automorphic representations.[17][18] Braid groups, which model the symmetries of strands intertwining in the plane, admit faithful linear representations, confirming their linearity. The Burau representation maps the n-strand braid group B_n into GL(n-1, ℤ[t, t^{-1}]), capturing the action on homology of punctured disks, though it is not faithful for n ≥ 5. The Lawrence-Krammer representation, however, provides a faithful embedding of B_n into GL((n choose 2), ℤ[q, t]), resolving the linearity question by explicitly constructing a discrete faithful action in a matrix group over a Laurent polynomial ring.[19][20] Nilpotent examples include the Heisenberg group over ℤ, defined as the set of 3×3 upper-triangular matrices with ones on the diagonal and integer off-diagonal entries: This group is generated by two elements, nilpotent of class 2, and embeds faithfully into GL(3, ℚ), illustrating how solvable infinite groups can arise linearly with polynomial growth rates.[21] A key structural theorem states that every finitely generated linear group over ℚ is residually finite, meaning it maps onto finite groups separating distinct elements. This property, due to Malcev, ensures effective algorithms for the word problem in such groups and underpins their embeddability into profinite completions; for instance, SL(n, ℤ) and the Heisenberg group over ℤ exemplify this residual finiteness through congruence quotients. The Tits alternative further dichotomizes these groups: a finitely generated linear group is either virtually solvable or contains a non-abelian free subgroup.[14][22]Geometric Examples
Fuchsian groups provide a fundamental class of linear groups arising from geometric actions on the hyperbolic plane. These are defined as discrete subgroups of the projective special linear group , which acts by orientation-preserving isometries on the upper half-plane model of hyperbolic 2-space.[23] The discreteness ensures a properly discontinuous action, allowing the quotient space to form a hyperbolic surface with finite or infinite area.[24] A prominent example is the modular group , which is generated by the transformations and , and serves as the fundamental group of the modular surface.[24] This linearity stems directly from the embedding in the linear group , modulo its center.[25] Kleinian groups extend this geometric perspective to three dimensions, consisting of discrete subgroups of , the group of orientation-preserving isometries of hyperbolic 3-space .[26] These groups act on the Riemann sphere at infinity, partitioning it into a limit set and a domain of discontinuity, with the quotient yielding hyperbolic 3-manifolds.[27] Their linearity follows from the representation as complex matrices in , up to scalars.[28] Examples include Schottky groups, generated by pairing disjoint disks on the sphere, which produce handlebodies as quotients.[26] Crystallographic groups illustrate linear actions in Euclidean geometry, where finite subgroups of the orthogonal group act as point groups on , preserving a lattice and generating the full discrete symmetry group of a crystal.[29] In three dimensions, there are 11 such finite linear groups in SO(3), corresponding to the chiral crystallographic point groups.[30][31] These actions are faithful linear representations, embedding the groups into .[29] Orientation-preserving isometries of compact surfaces of genus at least two yield linear fundamental groups through their action on the universal cover, the hyperbolic plane.[32] The deck transformation group is a Fuchsian subgroup of , providing a faithful linear representation of the surface group.[33] For instance, the fundamental group of the torus acts via affine transformations, but higher-genus cases embed into .[34] The uniformization theorem connects these examples to Riemann surfaces, asserting that every simply connected Riemann surface is conformally equivalent to the unit disk, complex plane, or sphere, with non-spherical cases admitting linear fundamental groups via Fuchsian actions on the disk.[35] For hyperbolic Riemann surfaces (genus ), the theorem implies the fundamental group is a discrete subgroup of , hence linear.[36] This uniformizes the geometry, embedding surface groups linearly into classical groups like .[37]Structural Properties
The Tits Alternative
The Tits alternative, established by Jacques Tits in 1972, resolves a key question in group theory analogous to the Burnside problem restricted to linear groups, determining whether finitely generated groups of bounded exponent are finite. Specifically, it asserts that every finitely generated subgroup of , where is any field, is either virtually solvable (i.e., contains a solvable subgroup of finite index) or contains a non-abelian free subgroup of rank 2.[15] This dichotomy highlights the structural rigidity of linear groups, contrasting with the more varied behaviors possible in general finitely generated groups.[15] The proof proceeds by analyzing the action of the group on projective space via the projective linear group , reducing the problem to the semisimple part of the Zariski closure of the group. Tits employs the ping-pong lemma, a combinatorial tool for detecting free subgroups in actions on trees or spaces: given a finite set of semisimple elements acting on the projective line with distinct attracting and repelling fixed points, sufficiently high powers of these elements play a "ping-pong" game, partitioning the space into disjoint fundamental domains whose preimages under the action ensure freeness.[15] This lemma applies after embedding into a suitable algebraic group and using field extensions to separate eigenvalues, guaranteeing the existence of such hyperbolic elements unless the group is virtually solvable.[38] Unipotent elements, which complicate the semisimple reduction, are handled separately: for a unipotent , there exists an integer such that the one-parameter subgroup generated by is connected in the Zariski topology, bounding its growth and ensuring it lies in a virtually nilpotent (hence virtually solvable) subgroup.[15] If the overall group growth is exponential—detectable via the subgroup's action—then the presence of elements violating polynomial growth bounds implies a free non-abelian subgroup, as virtually solvable linear groups exhibit only polynomial growth by the Milnor-Wolf theorem.[15] While the theorem holds over fields of characteristic zero without qualification, positive characteristic introduces nuances: the full linear group over an infinite algebraic extension of a finite field may fail to contain a non-abelian free subgroup yet not be virtually solvable, though finitely generated subgroups still satisfy the alternative.[38] In characteristic 2, additional care is needed in the ping-pong construction due to limited eigenvalue separations, but the dichotomy persists for finitely generated cases via modified arguments.[39]Residual Finiteness and Solvability
A fundamental property of finitely generated linear groups is their residual finiteness. Specifically, any finitely generated subgroup of is residually finite, as established by Mal'cev's theorem from 1940, which leverages the existence of torsion-free finite-index subgroups (provided by Selberg's lemma from 1960) to construct separating finite quotients. Similarly, finitely generated subgroups of for finite fields are residually finite by Mal'cev's theorem from 1940, which relies on the finiteness of matrix groups over finite fields to produce homomorphisms onto finite groups. These results were refined and extended by Wehrfritz in his 1973 monograph on infinite linear groups, providing detailed structural characterizations that confirm residual finiteness across broader classes of linear groups over rings of integers. The residual finiteness of linear groups implies algorithmic separability of elements, meaning that for any non-identity element, there exists a homomorphism to a finite group where its image is non-trivial, enabling effective decision procedures for the word problem and membership testing in such groups. This property also ensures the existence of surjective homomorphisms onto finite groups, allowing linear groups to be approximated by their finite quotients, which is crucial for computational group theory applications.[40] Linear groups with bounded Prüfer rank—where the Prüfer rank is the minimal number such that every finitely generated abelian quotient has rank at most —are virtually solvable, meaning they possess solvable subgroups of finite index. This follows from structural theorems classifying such groups as extensions of solvable groups by finite groups, limiting the complexity of their derived series.[41] For instance, is residually finite, as it admits finite quotients via reduction modulo primes, but it is not solvable due to the presence of non-abelian free subgroups in its structure. Arithmetic linear groups, such as those arising from , exhibit the congruence subgroup property, which guarantees residual finiteness through a family of finite-index normal subgroups called principal congruence subgroups. The principal congruence subgroup of level is defined as where is the identity matrix; this is the kernel of the natural reduction homomorphism , where the target is finite. The intersection of all over is trivial, ensuring that every non-identity element is detected in some finite quotient.Contrasting Non-Linear Groups
Key Examples of Non-Linear Groups
The Grigorchuk group, constructed by Rostislav Grigorchuk in 1980 as a group of automorphisms of the infinite binary rooted tree, provides a seminal example of an infinite torsion group that is non-linear. Every non-identity element has finite order dividing , yet the group is infinite, violating Burnside's theorem that torsion subgroups of linear groups over fields of characteristic zero are finite.[42] A proof of its non-linearity follows from its intermediate growth rate, which lies strictly between polynomial and exponential; finitely generated linear groups over such fields exhibit either polynomial or exponential growth, making intermediate growth incompatible with linearity.[43] This evasion of the Tits alternative—lacking both non-abelian free subgroups and virtually solvable quotients—further underscores its non-linear nature. Thompson's group , introduced by Richard Thompson in 1965, is another prominent example of an infinite non-linear group whose commutator subgroup is simple. Defined as the group of piecewise linear homeomorphisms of the unit interval with finitely many breakpoints at dyadic rationals and slopes powers of 2, lacks a faithful finite-dimensional linear representation. Its non-linearity arises from the Tits alternative, as contains no non-abelian free subgroup and has no solvable subgroup of finite index.[44] This structure makes it impossible for to embed into for any finite and field of characteristic zero. The direct product , where the superscript denotes the full (unrestricted) infinite direct product over countable indices, exemplifies a non-linear abelian torsion group of uncountable cardinality. This group is uncountable, with cardinality , as it consists of all sequences in for the first factor and similarly for . However, is countable for any finite , since its elements are matrices with rational entries from a countable set; thus, no faithful embedding is possible.[45] Certain Baumslag-Solitar groups, such as , serve as non-linear examples among one-relator groups. This group is not residually finite, admitting a surjective endomorphism that is not injective, as shown by the map sending and . Linear groups over fields of characteristic zero are residually finite by Mal'cev's theorem, precluding such embeddings.[44]Tests and Open Questions on Linearity
Determining whether a given finitely generated group is linear remains a challenging problem in group theory, with several criteria providing necessary conditions for linearity or evidence for non-linearity. One key obstruction arises from Kazhdan's property (T): infinite groups possessing property (T) cannot be virtually nilpotent, whereas certain classes of linear groups, such as solvable linear groups of bounded derived length, are virtually nilpotent by results on triangularization and unipotent radicals.[46] Thus, for groups in these classes, the presence of property (T) implies non-linearity if the group is infinite and not virtually nilpotent.[47] A prominent open question concerns the linearity of mapping class groups of closed orientable surfaces of genus . While the braid groups, which are isomorphic to the mapping class groups of punctured disks, were shown to be linear via a faithful representation into for appropriate , this result does not extend directly to higher-genus closed surfaces. For genus 2, linearity was established through an explicit faithful representation into a matrix group over .[48] However, for , no such faithful linear representation is known, and the question remains unresolved as of 2025.[49] Recent developments, such as classifications of low-dimensional representations up to dimension , have provided new insights into representations of mapping class groups but have not resolved the linearity question.[49] The Golod-Shafarevich inequality provides a powerful tool for establishing non-linearity in certain p-groups: it bounds the dimensions of cohomology groups such that if for a finite p-group , then infinite pro-p quotients exist, yielding infinite finitely generated p-groups of bounded exponent.[50] Such groups are non-linear, as finitely generated linear groups of bounded exponent are finite by Schur's theorem on periodic linear groups. Historically, counterexamples to the Burnside problem—finitely generated infinite groups of bounded exponent—have frequently been non-linear, as exemplified by the Golod-Shafarevich constructions, which produce infinite discrete torsion groups that violate the finiteness condition for linear groups.[50] The Grigorchuk group, another such counterexample, is also non-linear due to its growth properties incompatible with matrix groups.[51]Representation Theory
Faithful Representations
A faithful representation of a group is a homomorphism into the general linear group over a vector space , where the kernel is trivial, ensuring that distinct elements of act as distinct linear transformations. Constructing such representations is central to embedding groups into linear groups, allowing the application of linear algebra tools to study group structure. For finite groups, the minimal dimension of a faithful complex representation, known as the representation dimension , is achieved by taking the direct sum of a minimal set of irreducible representations whose common kernel is trivial. This dimension is at most for any finite group , except for certain 2-groups with elementary abelian center of order 8, where it can be larger.[9] For infinite groups, faithful finite-dimensional representations exist only if the group is linear, and such representations can often be constructed as inductive limits of faithful representations of finite subgroups or quotients. For instance, arithmetic groups like arise as inductive limits of finite quotients , each embedded faithfully into , yielding a faithful representation in the limit. Over specific fields like the rationals , faithful representations of finitely generated groups can be described using the rational canonical form, which decomposes the -module into cyclic components via companion matrices of invariant factors, providing a canonical basis for verifying injectivity.[52] Every countable group admits a faithful representation over via the left regular representation on the Hilbert space , where acts by translation on its basis of delta functions, but this is infinite-dimensional; linearity requires a finite-dimensional faithful representation, distinguishing linear groups from non-linear ones.[53] For Lie groups, a canonical example is the adjoint representation , where is the Lie algebra and the action is by conjugation, which is faithful precisely when is centerless (for connected ). Classical groups such as and provide standard faithful representations of their defining types. To assess or bound faithfulness in a representation , the character satisfies if and only if has all eigenvalues 1, which for irreducible representations implies by Schur's lemma (up to scalars, normalized by determinant 1 in special linear cases); thus, only for provides a necessary condition for faithfulness, with stricter bounds like for in unitary representations ensuring no non-trivial scalar actions.[54] with equality to implying in faithful cases.[55]Character Varieties and Deformations
In representation theory, the character variety associated to a finitely generated discrete group and a reductive algebraic group over is defined as the geometric invariant theory quotient , which parametrizes conjugacy classes of representations as a complex algebraic variety.[56] This space captures the moduli of representations up to the action of by simultaneous conjugation, with smooth points corresponding to irreducible representations where the stabilizer is the center of .[56] Infinitesimal deformations of a representation are studied through the Zariski tangent space at in the character variety, which is isomorphic to the first cohomology group , where is the Lie algebra of viewed as a -module via the adjoint action.[57] The tangent space to the conjugacy orbit at is given by , consisting of the centralizer of in . Near a smooth point , the local dimension of the character variety satisfies .[57] These tools apply to rigidity phenomena, such as Mostow's rigidity theorem, which states that for a finite-volume hyperbolic manifold of dimension at least three, the holonomy representation is unique up to conjugacy, implying that the corresponding point in the character variety is isolated with vanishing tangent space . This rigidity extends to deformations, showing that hyperbolic structures on such manifolds cannot be continuously deformed while preserving the fundamental group action. Advancements around 2020 include computational methods for determining equations defining low-dimensional character varieties, particularly for knot and link groups, using algebraic geometry software such as Macaulay2 to compute Gröbner bases and primary decomposition for -representations. These tools enable explicit calculations of irreducible components and ideal points in varieties for 3-manifold groups, facilitating the detection of essential surfaces and non-integral representations in specific examples like two-bridge knots.[58] More recent work as of 2025 continues to develop these computational approaches, for instance, exploring disc patterns to construct and visualize character varieties of knot groups.[59]Applications and Connections
Geometric and Topological Applications
Linear groups play a fundamental role in hyperbolic geometry, particularly through the action of the projective special linear group on the hyperbolic plane . This group acts as the group of orientation-preserving isometries of , enabling the study of Fuchsian groups, which are discrete subgroups of , and their quotients forming hyperbolic surfaces.[60] Such actions facilitate the classification of two-dimensional hyperbolic geometries and the computation of geometric invariants like volumes of ideal polyhedra via associated zeta functions. In particular, the Selberg zeta function, defined over the lengths of closed geodesics on these surfaces, encodes spectral data that relates to volume estimates through the prime geodesic theorem, providing asymptotic counts of geodesic lengths that inform hyperbolic volume computations.[61] In knot theory, linear groups arise in the representation theory of knot groups, which are fundamental groups of knot complements in . The Wirtinger presentation of a knot group allows for explicit embeddings of peripheral subgroups into , reflecting the holonomy of hyperbolic structures on the complement when the knot is hyperbolic.[62] This linearity captures essential topological features, such as the meridian and longitude, enabling the study of Dehn fillings and their geometric realizations. A representative example is the trefoil knot group, which, despite the trefoil being a torus knot with a Seifert fibered complement, admits faithful representations into that highlight its peripheral structure and ideal points in the representation space. The geometrization theorem, proved by Perelman in 2003, decomposes every closed orientable irreducible three-manifold into geometric pieces modeled on one of Thurston's eight geometries, each of whose fundamental groups is linear. This structure supports Thurston's conjecture that all fundamental groups of three-manifolds are linear—a question that remains open as of 2025—resolving long-standing questions about subgroup separability for specific classes and providing a pathway to computational topology via linear representations.[63][64] Recent developments connect linear groups to Floer homology through the study of representation spaces. Sheaf-theoretic constructions of -Floer homology for knots define invariants that detect properties of character varieties, linking symplectic geometry to the topology of representation spaces up to smooth components.[65] Advances since 2020, including localization spectral sequences in real Heegaard Floer homology, further refine obstructions to fillings and representations, enhancing the analysis of linear actions on knot and manifold invariants.[66]Physics and Other Interdisciplinary Links
In quantum mechanics, the special unitary group SU(2) acts as the universal double cover of the special orthogonal group SO(3), which parametrizes rotations in three-dimensional space. This covering homomorphism ensures that quantum states for particles with half-integer spin, such as fermions, transform correctly under rotations, as a 360-degree rotation corresponds to a -1 phase in the SU(2) representation, requiring a full 720-degree rotation to return to the identity. The isomorphism of their Lie algebras su(2) ≅ so(3) allows shared infinitesimal generators, but the topological distinction of SU(2) being simply connected resolves multi-valuedness issues in wave functions for spin-1/2 systems.[67][68] Classical groups, including linear subgroups like SO(3) and its covers, underpin physical symmetry operations in quantum systems ranging from angular momentum to particle interactions.[69] In crystallography, space groups extend finite linear groups—subgroups of GL(n, ℝ) restricted by crystallographic constraints—through semidirect products with lattice translations, capturing the full symmetry of periodic crystal structures. These extensions incorporate point group actions, where finite linear transformations (rotations and reflections) act on the translation subgroup, enabling classification of the 230 three-dimensional space groups. Non-symmorphic space groups further generalize this by including screw axes and glide planes, which couple linear rotations to fractional translations, as formalized in group cohomology for abelian extensions.[70][71] Linear groups connect to elliptic curve cryptography through representations that embed aspects of the curve's structure into matrix groups, particularly via the Tate module, which yields faithful homomorphisms from the absolute Galois group to GL(2, ℤ/ℓℤ) for prime ℓ. These linear representations facilitate analysis of the discrete logarithm problem's hardness in elliptic curve groups over finite fields and support advanced protocols, such as those involving isogenies, by providing a linear algebraic framework for point operations and security reductions.[72] In machine learning since 2020, group-equivariant neural networks incorporate linear representations of groups within GL(n, ℝ) to enforce inductive biases for symmetry-aware processing, enhancing efficiency in domains like molecular modeling and image analysis. These architectures decompose features into irreducible representations, ensuring the network's output respects the group's action, as in reductive Lie neurons that handle adjoint actions on Lie algebras for matrix-valued data. The core transformation law for equivariance (often termed covariance in this context) is expressed as where acts linearly on the input , denotes the representation on the input space, and is the output representation, guaranteeing consistent predictions under transformations like scalings or shears.[73]Computational Methods
Computational methods for linear groups leverage their matrix representations to enable efficient algorithms, particularly over finite fields. Membership testing in matrix groups, such as subgroups of GL(d, q), often employs the MeatAxe algorithm, a Monte Carlo method that tests irreducibility by randomizing elements and factoring characteristic polynomials, with complexity O(d^{3.5} log q).[74] For deterministic approaches in special linear groups SL(d, q), constructive recognition algorithms build standard generating sets via dimension reduction (e.g., embedding into SL(4, q) subgroups) and lifting, achieving polynomial time in d and log q.[75] Row reduction techniques generalize to express elements in terms of Steinberg generators for Lie-type groups, facilitating membership via solving linear systems over the field.[74] Finding generating sets for linear groups typically uses variants of the Schreier-Sims algorithm adapted to matrix actions on vector spaces, selecting base points like eigenvectors to construct a base and strong generating set (BSGS), which supports efficient orbit computations and stabilizer chains.[74] This process exploits the geometry of the natural module, bounding orbit sizes by (q^d - 1)/(q - 1), and is effective for small dimensions d.[76] Software systems like GAP and Magma provide robust implementations for these computations. In GAP, matrix groups over finite fields support BSGS construction via permutation representations or direct matrix operations, with functions likeGeneratorsOfGroup and MinimalGeneratingSet yielding concise generating sets.[77] Magma's BSGS procedure uses heuristics like eigenspace selection for base points and integrates composition trees for structural decomposition, enabling membership tests and order computations in polynomial time for classical groups.[76]
The complexity of linear group problems varies; while isomorphism for matrix groups over finite fields is solvable in polynomial time using canonical forms and BSGS, broader recognition tasks (e.g., classifying abstract groups via linear representations) relate to the graph isomorphism problem, which is not known to be NP-complete but suspected to be intermediate. Residual finiteness of linear groups enables these algorithms by approximating via finite quotients.[74]
Recent advances include quantum algorithms for computing representations of finite groups. The quantum representation framework encodes group generators as unitary circuits over finite-dimensional Hilbert spaces, using variational methods to satisfy relations, with applications to hidden subgroup problems via linear algebra over finite fields.[78]
For finite linear groups, irreducible representations are computed via character tables. In GAP, the Dixon-Schneider algorithm constructs the table by initializing with linear characters and splitting via class sum matrices, then derives representations from orthogonalities; for example, the character table of PSL(2,5) yields five irreducibles of degrees 1, 3, 3, 4, 5.[79][80]References
- https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_special_linear_group:SL(2,5)