Given a finitely generated group G, the set Hom(G,SL_2 C) inherits the structure of an algebraic variety R(G)called the "representation variety" of G. This algebraic variety is an invariant of G. Let G_{pt}=< a, b; a^p= b^t>, where p, t are integers greater than one. In this paper a formula is produced yielding the number of four dimensional irreducible components of the affine algebraic variety R(G_{pt}). A direct consequence of the main theorem of this paper is that if K is a torus knot, then its genus equals the number of four dimensional components of R(G_{pt}) corresponding to its knot group G_{pt}.
The focus of this article is to define the descriptively approximations in proximal relator spaces. Afterwards, descriptive approximately algebraic structures such as groupoids, semigroups and groups in digital images endowed with descriptive Efremovic proximity relations were introduced.
Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible t…
We consider word maps and word maps with constants on a simple algebraic group. We present results on the images of such maps, in particular, we prove a theorem on the dominance of general word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the corresponding representation variety.
This is a survey of two papers joint with A. Borisov and a paper joint with I. Spakulova. It is based on my lectures at the conference "Groups St. Andrews 2009", Bath (August 2009). We prove that almost all 1-related groups with at least 3 generators are virtually residually (finite p-)groups for almost all primes p, and coherent. The proof involves methods from combinatorial group theory (the congruence extension property of certain subgroups of free groups) algebraic geometry (dynamics of polynomial maps over finite and p-adic fields) and probability theory (convex hulls of Brownian bridges)…
We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.
A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic groups $G$ which are connected, closed and simple of rank at least $2$. Consequences are given concerning the representations of such subgroups on various $G$-modules: for example, with one exception, the conjugacy classes of irreducible simple connected subgroups of rank at least $2$ are determined by their composition factors on the adjoint module for $G$.
Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on {t_1,...,t_p, x_1,...,x_g, y_1,...y_g} which fix the element t_1t_2...t_p[x_1,y_1]...[x_g,y_g] and permute the set of conjugacy classes {[t_1],....,[t_p]}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labruère and Paris also gave …
Our main goal is to determine, under certain restrictions, the maximal closed connected subgroups of simple algebraic groups containing a regular torus. We call a torus regular if its centralizer is abelian. We also obtain some results of independent interest. In particular, we determine the irreducible representations of simple algebraic groups whose non-zero weights occur with multiplicity 1.
We show that for any positive integer $m\ge 1$, $m$-relator quotients of the modular group $M = PSL(2,\mathbb{Z})$ generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We also prove that such quotients are generically "essentially incompressible". By this we mean that their "absolute $T$-invariant", measuring the smallest size of any possible finite presentation of the group, is bounded below by a function which is almost linear in terms of the length of the given presentation. We compute the precise asymptotics of the number $I_m(n)$ of \emph{isomorphism types} of $m$-re…
This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is called irreducible if it lies in no proper parabolic subgroup of $G$. In this paper we complete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning t…
In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $ϕ$ be a non-trivial irreducible representation of $G$. Then the Jordan normal form of $ϕ(u)$ contains at most one non-trivial block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element and $\dim ϕ\leq 7$. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent eleme…
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$ and let $V$ be an irreducible rational $G$-module with highest weight $λ$. When $V$ is self-dual, a basic question to ask is whether $V$ has a non-degenerate $G$-invariant alternating bilinear form or a non-degenerate $G$-invariant quadratic form. If $p \neq 2$, the answer is well known and easily described in terms of $λ$. In the case where $p = 2$, we know that if $V$ is self-dual, it always has a non-degenerate $G$-invariant alternating bilinear form. However, determining when $V$ …
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups…
Let $H$ be a linear algebraic group whose connected component $G\neq 1$ is simple and $H/G$ is cyclic. We determine the irreducible projective representations $ϕ$ of $H$ such that $ϕ(G)$ is irreducible and $ϕ(h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $ϕ( \langle h\rangle)$ are of multiplicity 1. (Here $\langle h\rangle$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras associated to these parabolic subgroups are commutative, so the decomposition in irreducible components of the finite quasi-regular representations is given by the double cosets …
We show that a tensor product of irreducible, finite dimensional representations of a simple Lie algebra over a field of characteristic zero, determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers.
Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p \ge 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a non-trivial irreducible tensor-indecomposable $p$-restricted rational $KG$-module such that the restriction of $V$ to $H$ is irreducible. In this paper we classify all such triples $(G,H,V)$, where $H$ is a maximal closed disconnected positive-dimensional subgroup of $G$, and $H$ preserves a natural geometric structure on $W$.
Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V) of this form, where $V \ne W, W^{*}$ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible …
We study the spectra of non-regular semisimple elements in irreducible representations of simple algebraic groups. More precisely, we prove that if G is a simply connected simple linear algebraic group and f is a non-trivial irreducible representation of G in some GL(V) for which there exists a non-regular non-central semisimple element s in G such that f(s) has almost simple spectrum, then, with few exceptions, G is of classical type and dim V is minimal possible. Here the spectrum of a diagonalizable matrix is called simple if all eigenvalues are of multiplicity 1, and almost simple if at mo…
Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor indecomposable and rational. Assume that the restriction of $V$ to $H$ is irreducible. In this paper, we study the triples $(G,H,V)$ of this form when $G$ is a classical group and $H$ is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples $(G,H,V)$ to the case w…
Given a finitely generated (fg) group G, the set R(G) of homomorphisms from G to SL(2,C) inherits the structure of an algebraic variety known as the "representation variety" of G. This algebraic variety is an invariant of fg presentations of G. Call a group G parafree of rank n if it shares the lower central sequence with a free group of rank n, and if it is residually nilpotent. The deviation of a fg parafree group is the difference between the minimum possible number of generators of G and the rank of G. So parafree groups of deviation zero are actually just free groups. Parafree groups that…
Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $ϕ$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say that $s$ is strongly regular if $α(s)\neβ(s)$ for all distinct $T$-roots $α$ and $β$ of $G$. Our main result states that if all but one of the eigenvalues of $ϕ(s)$ are of multiplicity 1 then, with a few specified exceptions, $s$ is strongly regular. This can be viewed as an extension of our earlier result saying that under the same hypotheses, $s$ must be regul…
For $G$ an algebraic group of type $A_l$ over an algebraically closed field of characteristic $p$, we determine all irreducible rational representations of $G$ in defining characteristic with dimensions $\le (l+1)^s$ for $s = 3, 4$, provided that $l > 18$, $l > 35$ respectively. We also give explicit descriptions of the corresponding modules for $s = 3$.
Consider an algebraic semigroup $S$ and its closed subscheme of idempotents, $E(S)$. When $S$ is commutative, we show that $E(S)$ is finite and reduced; if in addition $S$ is irreducible, then $E(S)$ is contained in a smallest closed irreducible subsemigroup of $S$, and this subsemigroup is an affine toric variety. It follows that $E(S)$ (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when $S$ is an irreducible algebraic monoid, we show that $E(S)$ is smooth, and its connected components are conjugacy classes of the unit group.
Recently Dicks-Linnell determined the $L^2$-Betti numbers of the orientable surface-plus-one-relation groups, and their arguments involved some results that were obtained topologically by Hempel and Howie. Using algebraic arguments, we now extend all these results of Hempel and Howie to a larger class of two-relator groups, and we then apply the extended results to determine the $L^2$-Betti numbers of the non-orientable surface-plus-one-relation groups.
We use tilting modules to study the structure of the tensor product of two simple modules for the algebraic group $\SL_2$, in positive characteristic, obtaining a twisted tensor product theorem for its indecomposable direct summands. Various other related results are obtained, and numerous examples are computed.
A complete description of the Alexander modules of knotted $n$-manifolds in the sphere $S^{n+2}$, $n\geq 2$, and irreducible Hurwitz curves is given. This description is applied to investigate properties of the first homology groups of cyclic coverings of the sphere $S^{n+2}$ and the projective complex plane $\mathbb C\mathbb P^2$ branched respectively alone knotted $n$-manifolds and along irreducible Hurwitz (in particular, algebraic) curves.
We construct a moduli space for the connected subgroups of an algebraic group and the corresponding universal family. Morphisms from an algebraic variety to this moduli space correspond to flat families of connected algebraic subgroups parametrised by this variety. This moduli space is obtained by gluing together infinitely many irreducible projective varieties of bounded dimension along closed subvarieties. Regarding families of non-connected subgroups of an algebraic group, we show that, given sich a family, the corresponding family of identity components is an irreducible component of the f…
We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
The idea that the cohomology of finite groups might be fruitfully approached via the cohomology of ambient semisimple algebraic groups was first shown to be viable in the papers [CPS75] and [CPSvdK77]. The second paper introduced, through a limiting process, the notion of generic cohomology, as an intermediary between finite Chevalley group and algebraic group cohomology. The present paper shows that, for irreducible modules as coefficients, the limits can be eliminated in all but finitely many cases. These exceptional cases depend only on the root system and cohomological degree. In fact, we …
We establish an irreducibility property for the characters of finite dimensional, irreducible representations of simple Lie algebras (or simple algebraic groups) over the complex numbers, i.e., that the characters of irreducible representations are irreducible after dividing out by (generalized) Weyl denominator type factors. For $SL(r)$ the irreducibility result is the following: let $λ=(a_1\geq a_2\geq ... a_{r-1}\geq 0)$ be the highest weight of an irreducible rational representation $V_λ$ of $SL(r)$. Assume that the integers $a_1+r-1, ~a_2+r-2,..., a_{r-1}+1$ are relatively prime. Then the…
