The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

Über die Glattheit von Quotientenabbildungen

Let G be a semisimple algebraic group acting on a factorial Gorenstein algebra S. Let X:=Spec S, Y:=Spec S^G and π:X→Y be the quotient map. The main results are:

1. Let x be a smooth point of X whose orbit has maximal dimension and such that π(x) is a smooth point of Y. Then π is smooth at x.
2. Let S be positively graded and let χ_S(t) be its generating function which is a rational function. Then: deg χ_S≦deg \(X_{S^G } \) .

     

Space of operators acting from one banach lattice to another

In this note we construct a pair of Banach lattices X and Y, which have the following properties:

1. X is not order isomorphic to an AL-space,
2. Y is not order isomorphic to an AM-space,
3. for any continuous linear operator T:X → Y there exists a modulus ¦T¦: X → Y.

This example refutes the conjecture of Cartwright-Lotz, saying that the negation of at least one of the conditions a) or b) is necessary for the validity of c).     

Standard completions for quasiordered sets

A standard completion for a quasiordered set Q is a closure system whose point closures are the principal ideals of Q. We characterize the following types of standard completions by means of their closure operators:

1. V-distributive completions,
2. Completely distributive completions,
3. A-completions (i.e. standard completions which are completely distributive algebraic lattices),
4. Boolean completions.

Moreover, completely distributive completions are described by certain idempotent relations, and the A-completions are shown to be in one-to-one correspondence with the join-dense subsets of Q. If a pseudocomplemented meet-semilattice Q has a Boolean completion ?, then Q must be a Boolean lattice and ? its MacNeille completion.     

On projective planes admitting elations and homologies

We consider projective planes Π of ordern with abelian collineation group Γ of ordern(n?1) which is generated by (A, m)-elations and (B, l)-homologies wherem =AB andA εl. We prove

1. Ifn is even thenn=2^e and the Sylow 2-subgroup of Γ is elementary abelian.
2. Ifn is odd then the Sylow 2-subgroup of Γ is cyclic.
3. Ifn is a prime then Π is Desarguesian.
4. Ifn is not a square thenn is a prime power.

     

Coherence classes of ideals of normal lattices with applications to C(X)

Given a topological space X, Jenkins and McKnight have shown how ideals of the ring C(X) are partitioned into equivalence classes — called coherence classes — defined by declaring ideals to be equivalent if their pure parts are identical. In this paper we consider a similar partitioning of the lattice of ideals of a normal bounded distributive lattice. We then apply results obtained herein to augment some of those of Jenkins and McKnight. In particular, for Tychonoff spaces, new results include the following:

1. all members of any coherence class have the same annihilator
2. every ideal is alone in its coherence class if and only if the space is a P-space.

     

The deformation theory of representations of fundamental groups of compact Kähler manifolds

Let Γ be the fundamental group of a compact Kähler manifold M and let G be a real algebraic Lie group. Let ?(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ?(Γ, G) it is shown that there exists a neighborhood of ρ in ?(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space \(Z^1 (\Gamma ,g_{Ad\rho } )\) of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in \(H^2 (\Gamma ,g_{Ad\rho } )\) . We prove that ?(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ?(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

Laguerre-Ebenen mit Symmetrien an Punktepaaren und Zykeln

Let ∞ be a point of a Laguerre plane, such that

1. For any cycle containing ∞ there exists an automorphism of order 2 whose set of fixed points is exactly z.
2. For any point X, not parallel to ∞, there exists an automorphism of order 2 whose set of fixed points is exactly {∞,X}.

Then the give Laguerre plane is a Miquelian one of characteristik ≠ 2.     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.     

States on bounded commutative residuated lattices

We define states on bounded commutative residuated lattices and consider their property. We show that, for a bounded commutative residuated lattice X,

1. If s is a state, then X/ker(s) is an MV-algebra.
2. If s is a state-morphism, then X/ker(s) is a linearly ordered locally finite MV-algebra.

Moreover we show that for a state s on X, the following statements are equivalent:

1. s is a state-morphism on X.
2. ker(s) is a maximal filter of X.
3. s is extremal on X.

     

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon–Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free group of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E_Γ of F, and the main theorem of Dowdall–Taylor [DT14] states that in this situation E_Γ is hyperbolic if and only if Γ is purely atoroidal.     

Cyclic algebras,complete fields,and crossed products

Theorem 1

Let q=char(k). Let M be a subfield of D which is Galois over K of degree m with Galois group H.

1. If q/m then H has a normal q-Sylow subgroup.
2. Iq q ? m then H is an abelian group with one or two generators, an extension of a cyclic group by a cyclic group of order e where k contains a primitive e-th root of unity.

Letk(X) be the generic division ring overk of indexn as defined by Amitsur.

Theorem 2

If n is divisible by the square of a prime p≠char(k) and k does not contain a primitive p-th root of unity, then k(X) is not a crossed product.     

Topologies on products of partially ordered sets III: Order convergence and order topology

This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are:

1. Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5).
2. For any product of posets, the projections are open and continuous with respect to the order topologies (2.1).
3. A productL of chainsL _i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology onL agrees with the product topology (2.7).
4. If (L _i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8).
5. LetP ₁ be a poset with topological order convergence and locally compact order topology. Then for any posetP ₂, the order topology ofP ₁?P ₂ coincides with the product topology (2.10).
6. A latticeL which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology ofL?L is the product topology (2.15).

Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.     

On a permutablity problem for groups

Letm, n be positive integers. We denote byR(m, n) (respectivelyP(m, n)) the class of all groupsG such that, for everyn subsetsX ₁, X₂, . . .,X _n of sizem ofG there exits a non-identity permutation σ such that $X_1 X_2 ...X_n \cap X_{\sigma (1)} X_{\sigma (2)} ...X_{\sigma (n)} \ne \not 0$ (respectively X₁X₂ . . .X _n = X_σ(1)X{σ(2)} . . . X{gs(n)}). Let G be a non-abelian group. In this paper we prove that

1. G ∈ P(2,3) if and only ifG isomorphic to S₃, whereS _n is the symmetric group onn letters.
2. G ∈ R(2, 2) if and only if¦G¦ ≤ 8.
3. IfG is finite, thenG ∈ R(3, 2) if and only if¦G¦ ≤ 14 orG is isomorphic to one of the following: SmallGroup(16,i), i ∈ {3, 4, 6, 11, 12, 13}, SmallGroup(32,49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of orderm in the GAP [13] library.

     

On lattice properties of the composition operator

The vector space £_b(E) of all order bounded linear operators on a Dedekind complete Riesz space E is both a Riesz space and an algebra. This note investigates the degree of compatibility between the algebraic and lattice structures of £_b(E). Two of the main results are the following:

1. An operator on a Banach lattice with an order continuous norm factors through the lattice operations if and only if it is an interval preserving Riesz homotnorphism.
2. A Dedekind complete Banach lattice E has an order continuous norm if and only if 0≤T_n ↑ T in £_b(E) implies T _n ² ↑ T².

     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces

We compute the quasi-isometry group of an irreducible nonuniform lattice in a semisimple Lie group with finite center and no rank one factors, and show that any two such lattices are quasi-isometric if and only if they are commensurable up to conjugation.

     

On a conjecture about automorphisms of finite p-groups

Let G be a nonabelian finite p-group. A longstanding conjecture asserts that G admits a noninner automorphism of order p. In this paper, we prove that if G satisfies one of the following conditions

1. ${\mathrm{rank}(G'\cap Z(G))\neq \mathrm{rank}(Z(G))}$
2. ${\frac{Z_{2}(G)}{Z(G)}}$ is cyclic
3. C _G (Z(Φ(G))) = Φ(G) and ${\frac{Z_{2}(G)\cap Z(\Phi(G))}{Z(G)} }$ is not elementary abelian of rank rs, where r = d(G) and s = rank (Z(G)),

then G has a noninner central automorphism of order p which fixes Φ(G) elementwise.     

Symmetric differentials, Kähler groups and ball quotients

While Margulis’ superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real Lie groups, almost nothing is known concerning the representation theory of complex hyperbolic lattices. The main result of this paper (Theorem 1.3) is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices. It relies on the following two ingredients:

• Theorem 1.6 showing that the representations of the topological fundamental group of a compact Kähler manifold X are controlled by the global symmetric differentials on X.
• An arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms, in particular deep results of Clozel on base change (Theorem 1.11).