Varietal isologism and covering groups

In this paper it will be shown that any two \bf\cal V-covering groups of a given group are V\bf\cal V-isologic with respect to the variety V\bf\cal V, which is a vast generalization of a result in B. Huppert (1967) and R. L. Griess JR (1973). We also give a criterion of existence of V\bf\cal V-covering groups for a V\bf\cal V-perfect group, and show that every automorphism of a given V\bf\cal V-perfect group G can be extended to an automorphism of the V\bf\cal V-covering G^* say, of G, this generalizes a result of J. L. Alperin and D. Gorenstein (1966), in the abelian variety.     

A combinatorial condition on a certain variety of groups

Let n be an integer and B_n \mathcal B_n be the variety defined by the law [xⁿ,y][x,yⁿ]^-1 = 1.¶ Let B_n^* \mathcal B_n^* be the class of groups in which for any infinite subsets X, Y there exist x ? X x \in X and y ? Y y \in Y such that [xⁿ,y][x,yⁿ]^-1 = 1. For $ n \in {\pm 2, 3\} $ n \in {\pm 2, 3\} we prove that¶ B_n^* = B_n èF \mathcal B_n^* = \mathcal B_n \cup \mathcal F , F \mathcal F being the class of finite groups. Also for $ n \in {- 3, 4\} $ n \in {- 3, 4\} and an infinite group G which has finitely many elements of order 2 or 3 we prove that G ? B_n^* G \in \mathcal B_n^* if and only if G ? B_n G \in \mathcal B_n .     

Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold

We show that for every closed Riemannian manifold X there exists a positive number¶ $ \varepsilon_0 > 0 $ \varepsilon_0 > 0 such that for all 0< e\leqq e₀ \varepsilon \leqq \varepsilon_0 there exists some¶ $ \delta > 0 $ \delta > 0 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ d \delta the geometric e \varepsilon -complex |Y_e| |Y_\varepsilon| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann [4].     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

A characterization of infinite 3-abelian groups

In this note we prove that every infinite group G is 3-abelian (i.e. (ab)³ = a³b³ for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ? Xx\in X and y ? Yy\in Y such that (xy)³ = x³y³.     

How Tight Is the Bollobás-Komlós Conjecture?

The bipartite case of the Bollobás and Komlós conjecture states that for every j₀, %>0 there is an !=!(j₀, %) >0 such that the following statement holds: If G is any graph with minimum degree at least n$\displaystyle {n \over 2}+%n then G contains as subgraphs all n vertex bipartite graphs, H, satisfying¶H)hj₀ \quad {\rm and} \quad b(H)h!n.$j (H)hj₀ \quad {\rm and} \quad b(H)h!n.¶Here b(H), the bandwidth of H, is the smallest b such that the vertices of H can be ordered as v₁, …, v_n such that v_i~_Hv_j implies |imj|hb.¶ This conjecture has been proved in [1]. Answering a question of E. Szemerédi [6] we show that this conjecture is tight in the sense that as %̂ then !̂. More precisely, we show that for any 0 such that that !(j₀, %)Д %.     

Nonamenable products are not treeable

LetX andY be infinite graphs such that the automorphism group ofX is nonamenable and the automorphism group ofY has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct productX×Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product has infinitely many connected components almost surely. Research partially supported by NSF grant DMS-9803597.     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.     

Invariant percolation and harmonic Dirichlet functions

The main goal of this paper is to answer Question 1.10 and settle Conjecture 1.11 of Benjamini–Lyons–Schramm [BenLS] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the random-cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for random-cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of ℓ² Betti numbers. Received: May 2004 Revised: March 2005 Accepted: May 2005     

On the selfsimilar solutions of a diffusion convection equation

This paper is concerned with the equation¶¶ div(| ?u| ^p-2?u)+e| ?U| ^q+bx?U+aU=0, for  x ? \mathbbR^N div(| \nabla u| ^{p-2}\nabla u)+\varepsilon \left| \nabla U\right| ^q+\beta x\nabla U+\alpha U=0,{\rm \ for}\;x\in \mathbb{R}^N ¶¶ where $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 $ p>2,\;q\geq 1,\;N\geq 1, \quad\varepsilon =\pm 1 and a,b, m \alpha ,\beta, \mu are positive parameters. We study the existence, uniqueness of radial solutions u(r). Also, qualitative behavior of u(r) are presented.     

Behaviour of doubly connected minimal surfaces at the edges of the support surface

This paper is the second part of our investigations on doubly connected minimal surfaces which are stationary in a boundary configuration (G, S) (\Gamma, S) in \Bbb R ³ \Bbb R ^3 . The support surface S is a vertical cylinder above a simple closed polygon P(S) P(S) in the x,y-plane. The surrounding Jordan curve G \Gamma is chosen as a generalized graph above its convex projection curve P(G) P(\Gamma) . In [23] we have proved the existence of nonparametric minimal surfaces X of annulus type spanning such boundary configurations. We study the behaviour of these minimal surfaces at the edges of the support surface S. In particular we discuss the phenomenon of edge-creeping, i. e. the fact that the free trace of X may attach to an edge of S in a full interval. We prove that a solution X cuts any intruding edge of S perpendicularly. On the other hand, we derive a condition which forces X to exhibit the edge-creeping behaviour. Depending on the symmetries of (G, S) (\Gamma, S) we give bounds on the number of edges where edge-creeping occurs. Let (x,y,Z (x,y)) (x,y,\hbox {Z} (x,y)) for (x,y) ? G (x,y)\in G be the nonparametric representation of X. Then at every vertex Q of P(S) P(S) the radial limits of Z from all directions in G exist.     

Cellular automata, quasigroups and symmetries

Summary. We investigate cellular automata (CA) with a local rule f: G² ? G \phi : G^2 \rightarrow G , where the local rule defines a quasigroup structure (Latin square) on the finite set G. If the quasigroup is semisymmetric or totally symmetric, some top-down equilateral triangular subsets of the CA-orbits, the so-called \triangledown \triangledown -configurations, exhibit certain symmetries. The most interesting symmetries are the rotational and the total (dihedral) symmetries, which may be considered in conjunction with certain automorphisms.¶We first explore the conditions for quasigroups to be symmetric (or for local CA-rules to allow symmetric \triangledown \triangledown -configurations), and how to construct symmetric quasigroups by prolongation, i.e., by steadily increasing the order of the quasigroup, thereby conserving the symmetry. Then we study rotationally or totally symmetric \triangledown \triangledown -configurations. We begin with the existence of symmetric \triangledown \triangledown -configurations of arbitrary size N and N o 0,1 mod 3 N \equiv 0,1\,{\rm mod}\,3 , and show that symmetric \triangledown \triangledown -configurations of size N o 2 mod 3 N \equiv 2\,{\rm mod}\,3 exist under mild conditions on J. We present explicit formulas for the number of distinct symmetric \triangledown \triangledown -configurations. By studying the combined group action of the dihedral (or rotational) group and the automorphism group of the quasigroup G on the \triangledown \triangledown -configurations of size N, we are able to classify and count the number of different equivalence classes of \triangledown \triangledown -configurations.     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.     

Orbit nonproper actions on Lorentz manifolds

If a topological group G acts on a topological space X, then we say that the action is orbit nonproper provided that, for some x ? X x \in X , the orbit map g ? gx : G ? X g \mapsto gx : G \to X is nonproper. We consider the problem of classifying the connected, simply connected real Lie groups G such that G admits a locally faithful, orbit nonproper action on a connected Lorentz manifold. In this paper, we describe three collections of groups such that G admits such an action iff G is in one of the three collections. In an earlier paper, we effectively described the first collection. In yet another paper, we describe effectively those groups in the second collection which are not contained in the union of the first and third. Finally, in another paper, we describe effectively the third collection.     

Distinguishing graphs with intermediate growth

A graph G is said to be 2-distinguishable if there is a 2-labeling of its vertices which is not preserved by any nontrivial automorphism of G. We show that every locally finte graph with infinite nite motion and growth at most \(\mathcal{O}\left( {2^{(1 - \varepsilon )\tfrac{{\sqrt n }}{2}} } \right)\) is 2-distinguishable. Infinite motion means that every automorphism moves infinitely many vertices and growth refers to the cardinality of balls of radius n.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

Wilson's functional equation on P3-groups

Summary. Consider Wilson's functional equation¶¶f(xy) + f(xy^-1) = 2f(f)g(y) f(xy) + f(xy^{-1}) = 2f(f)g(y) , for f,g : G ? K f,g : G \to K ¶where G is a group and K a field with char K 1 2 {\rm char}\, K\ne 2 .¶Aczél, Chung and Ng in 1989 have solved Wilson's equation, assuming that the function g satisfies Kannappan's condition g(xyz) = g(xzy) and f(xy) = f(yx) for all x,y,z ? G x,y,z\in G .¶In the present paper we obtain the general solution of Wilson's equation when G is a P₃-group and we show that there exist solutions different of those obtained by Aczél, Chung and Ng.¶A group G is said to be a P₃-group if the commutator subgroup G' of G, generated by all commutators [x,y] := x^-1y^-1xy, has the order one or two.     

Solutions to five problems on tensor products of lattices and related matters

The notion of a capped tensor product, introduced by G. Grätzer and the author, provides a convenient framework for the study of tensor products of lattices that makes it possible to extend many results from the finite case to the infinite case. In this paper, we answer several open questions about tensor products of lattices. Among the results that we obtain are the following:¶¶Theorem 2. Let A be a lattice with zero. If A ?B A \oplus B is a lattice for every lattice L with zero, then A is locally finite and A ?B A \oplus B is a capped tensor product for every lattice L with zero.¶¶Theorem 5. There exists an infinite, three-generated, 2-modular lattice K with zero such that K ?K K \oplus K is a capped tensor product.¶¶Here, 2-modularity is a weaker identity than modularity, introduced earlier by G. Grätzer and the author.