On graded quotient modules of mapping class groups of surfaces

Let Γ _{g, n} be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ _{g, n} has a natural weight filtration {Γ_{g, n}(m)} _m≥1. Each graded quotient gr ^m Γ _{g, n} ⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S _n naturally acts. In this paper, we have determined the Sp(2g, ℚ)×S _n module structure of gr ^m Γ _{g, n} ⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr ^m Γ _{g, n} ⊗ ℚ by constructing explicitly elements in these modules.     

Adjacency preservers,symmetric matrices,and cores

It is shown that the graph Γ _n that has the set of all n×n symmetric matrices over a finite field as the vertex set, with two matrices being adjacent if and only if the rank of their difference equals one, is a core if n≥3. Eigenvalues of the graph Γ _n are calculated as well.     

Deformations of complex structures on Γ/SL 2(C)

LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL ₂(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston that this is true for all cocompact lattices inSL(2, C)). We also show thatG acts trivially on the coherent cohomology groupsH ⁱ(Γ/G, O) for anyi≥0.     

Structure of the automorphism group for partially commutative class two nilpotent groups

Let R be a ring, which is either a ring of integers or a field of zero characteristic. For every finite graph Γ, we construct an R-arithmetic linear group H(Γ). The group H(Γ) is realized as the factor automorphism group of a partially commutative class two nilpotent R-group G _Γ. Also we describe the structure of the entire automorphism group of a partially commutative nilpotent R-group of class two.     

Polynomial representation growth and the congruence subgroup problem

Let Γ be anS-arithmetic group in a semisimple group. We show that if Γ has the congruence subgroup property then the number of isomorphism classes of irreducible complexn-dimensional characters of Γ is polynomially bounded. In characteristic zero, the converse is also true. We conjecture that the converse also holds in positive characteristic, and we prove some partial results in this direction. Dedicated to Andy Magid on the occasion of his sixtieth birthday     

Verifying Huppert’s Conjecture for <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups ² G ₂(q ²) for q ² = 3^2m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.     

On the outer automorphism group of a hyperbolic group

LetG be a one-ended, word-hyperbolic group. Let Γ be an irreducible lattice in a connected semi-simple Lie group of rank at least 2. Ifh: Γ→Out(G) is a homomorphism, then Im(h) is finite. Dedicated to Professor Takushiro Ochiai for his sixtieth birthday     

Boundaries of random walks on graphs and groups with infinitely many ends

Consider an irreducible random walk {Z _n} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Z _n} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Z _n} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Z _n} converges to this end. If {Z _n} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.     

A low-memory algorithm for finding short product representations in finite groups

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-ρ approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log₂ n, where n = #G and d ≥ 2 is a constant, we find that its expected running time is O(?n log n){O(\sqrt{n}\,{\rm log}\,n)} group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.     

Direct products of cyclic semigroups an unions of groups

We consider direct productsS×U_eεE G _e=S ₁×…×S _n × U_eεE G _e of non-group finite cyclic semigroupsS _i, 1 ≤i ≤n, and finite unions of finite groups U_eεE G _e We prove that if such a semigroup is isomorphic to another of the same form, sayT×U _fεF H _f =T ₁×…×U _fεF H _f , whereT _j are non-group cyclic semigroups, 1≤j≤l, and U _fεF H _f is a union of groups, thenS is isomorphic toT and U_eεE G _e is isomorphic to U_fεF H _f . We then determine when a finite semigroup has such a decomposition and show how the direct factors can be found.     

On geometric density of Hecke eigenvalues for certain cusp forms

In this paper we prove certain density results for Hecke eigenvalues as well as we give estimates on the length of modules for Hecke algebra acting on the cusp forms constructed out of Poincaré series for a semisimple group G over a number field k. The cusp forms discusses here are taken from Muić (Math Ann 343:207–227, 2009) and they generalize usual cuspidal modular forms S _k (Γ) of weight k ≥ 3 for a Fuchsian group Γ (Muić, in On the cuspidal modular forms for the Fuchsian groups of the first kind).     

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon. It follows that any infinite simple group of finite Morley rank with a spherical MoufangBN-pair of Tits rank 2 is eitherPSL ₃(K),PSp ₄(K) orG ₂(K) for some algebraically closed fieldK. Spherical irreducible buildings of Tits rank ≥ 3 are uniquely determined by their rank 2 residues (i.e. polygons). Using Theorem A we show Theorem B:If G is an infinite simple group of finite Morley rank with a spherical Moufang BN-pair of Tits rank ≥ 2, then G is (interpretably) isomorphic to a simple algebraic group over an algebraically closed field. Theorem C:Let K be an infinite field, and let G(K) denote the group of K-rational points of an isotropic adjoint absolutely simple K-algebraic group G of K-rank ≥ 2. Then G(K) has finite Morley rank if and only if the field K is algebraically closed. We also obtain a result aboutBN-pairs in splitK-algebraic groups: such aBN-pair always contains the root groups. Furthermore, we give a proof that the sets of points, lines and flags of any ℵ₁-categorical polygon have Morley degree 1. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Supported by the Minerva Foundation (Germany). Research Director at the Fund for Scientific Research-Flanders (Belgium).     

Characterization of $ {\mathbb{A}_{16}} $ by a noncommuting graph

Let G be a finite non-Abelian group. We define a graph Γ _G ; called the noncommuting graph of G; with a vertex set G − Z(G) such that two vertices x and y are adjacent if and only if xy ≠ yx: Abdollahi, Akbari, and Maimani put forward the following conjecture (the AAM conjecture): If S is a finite non-Abelian simple group and G is a group such that Γ _S ≅ Γ _G ; then S ≅ G: It is still unknown if this conjecture holds for all simple finite groups with connected prime graph except \mathbbA₁₀ {\mathbb{A}_{10}} , L ₄(8), L ₄(4), and U ₄(4). In this paper, we prove that if \mathbbA₁₆ {\mathbb{A}_{16}} denotes the alternating group of degree 16; then, for any finite group G; the graph isomorphism G_\mathbbA16 @ G_G {\Gamma_{{\mathbb{A}_{16}}}} \cong {\Gamma_G} implies that \mathbbA₁₆ @ G {\mathbb{A}_{16}} \cong G .     

Subgroups Of Direct Products Of Elementarily Free Groups

The structure of groups having the same elementary theory as free groups is now known: they and their finitely generated subgroups form a prescribed subclass of the hyperbolic limit groups. We prove that if G ₁,...,G _n are in then a subgroup Γ ⊂ G ₁ × … × G _n is of type FP _n if and only if Γ is itself, up to finite index, the direct product of at most n groups from . This provides a partial answer to a question of Sela. This work was supported in part by Franco–British Alliance project PN 05.004. The first author is also supported by an EPSRC Senior Fellowship and a Royal Society Wolfson Research Merit Award. Received: July 2005 Accepted: April 2006     

Pseudo-coefficients des séries <Emphasis Type="Italic">κ</Emphasis>-discrètes de GL(<Emphasis Type="Italic">n,F</Emphasis>)

Let F be a non-Archimedean locally compact field, n be an integer ≥ 2, κ be a character of F ^× of finite order dividing n, and G be the group GL(n, F). We prove that if π is an irreducible κ-discrete series of G, then there exist some pseudo-coefficients for the twisted character trace(π ∘ A) where A denotes a non-zero intertwining operator between π ⊗ (κ ∘ det) and π.     

Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S ≤ G ≤ Aut(S) for a finite simple group S. More generally, we show that if G is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups. If the alternating group A ₇ is involved in G, the structure of G can be further restricted.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Minimal generating reflexive lattices of projections in finite von Neumann algebras

We show that the reflexive lattice generated by a double triangle lattice of projections in a finite von Neumann algebra is topologically homeomorphic to the two-dimensional sphere S ² (plus two distinct points corresponding to zero and I). Furthermore, such a reflexive lattice is in general minimally generating for the von Neumann algebra it generates. As an application, we show that if a reflexive lattice F{\mathcal F} generates the algebra M_n(\mathbb C){M_n(\mathbb C)} of all n × n complex matrices, for some n ≥ 3, then F\{0,I}{\mathcal F\setminus\{0,I\}} is connected if and only if it is homeomorphic to S ².