On the virtual residual p-finiteness of the free product of polycyclic groups with normal amalgamated subgroups

Let G be the free product of polycyclic groups A and B with normal amalgamated subgroups H and K. We prove that, for any prime p, the group G is a virtually residually p-finite group.     

On regular graphs,V

Let Γ₃ be an infinite regular tree of valence 3. There exist subgroups B of Aut (Γ₃) which are 5-regular on Γ₃, i.e., sharply transitive on the set of 5-arcs of Γ₃. We prove that any two such subgroups are conjugate in Aut (Γ₃). The pair (Γ₃, B) is a universal 5-regular action in the sense that if (G, A) is a pair consisting of a cubical graph G and a 5-regular subgroup A of automorphisms of G then (G, A) can be “covered” by (Γ₃, B) in a certain natural way.     

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A_Γ corresponding to an arbitrary Brauer tree Γ of type (m,e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A_S, whose tree is a star with the exceptional vertex in the middle, to A_Γ. The grading on A_S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute , the group of outer automorphisms that fix the isomorphism classes of simple A_Γ-modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A_Γ up to graded Morita equivalence and rescaling.     

Uniform embeddability and exactness of free products

Let A and B be countable discrete groups and let Γ=A∗B be their free product. We show that if both A and B are uniformly embeddable in a Hilbert space then so is Γ. We give two different proofs: the first directly constructs a uniform embedding of Γ from uniform embeddings of A and B; the second works without change to show that if both A and B are exact then so is Γ.     

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.      

Random mappings of scaled graphs

Given a connected finite graph Γ with a fixed base point O and some graph G with a based point we study random 1-Lipschitz maps of a scaled Γ into G. We are mostly interested in the case where G is a Cayley graph of some finitely generated group, where the construction does not depend on the choice of base points. A particular case of Γ being a graph on two vertices and one edge corresponds to the random walk on G, and the case where Γ is a graph on two vertices and two edges joining them corresponds to Brownian bridge in G. We show, that unlike in the case ${G=\mathbb Z^d}$ , the asymptotic behavior of a random scaled mapping of Γ into G may differ significantly from the asymptotic behavior of random walks or random loops in G. In particular, we show that this occurs when G is a free non-Abelian group. Also we consider the case when G is a wreath product of ${\mathbb Z}$ with a finite group. To treat this case we prove new estimates for transition probabilities in such wreath products. For any group G generated by a finite set S we define a functor E from category of finite connected graphs to the category of equivalence relations on such graphs. Given a finite connected graph Γ, the value E _G,S (Γ) can be viewed as an asymptotic invariant of G.     

The generating graph of some monolithic groups

For a finite group G let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. We look for conditions on the positive integer m that ensure that Γ(G) contains a Hamiltonian cycle when G=S?Cm is the wreath product of a finite simple group S and a cyclic group of order m.     

On the automorphism group of a tree

It is shown that H = Γ(T)_v is normal in G = Γ(T_v) for any tree T and any vertex v, if and only if, for all vertices u in the neighborhood N of v, the set of images of u under G is either contained in N or has precisely the vertex u in common with N and every vertex in the set of images is fixed by H. Further, if S is the smallest normal subgroup of G containing H then $\text{G}\text{S}$ is the direct product of the wreath products of various symmetric groups around groups of order 1 or 2. The degrees of the symmetric groups involved depend on the numbers of isomorphic components of T_v and the structure of such components.     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

KK ‐theory of amalgamated free products of C *‐algebras

J. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK ‐groups of the amalgamated free product A _{1 ?? B} A ₂ to the KK ‐groups of A ₁, A ₂ and B. First we establish automatic existence of strict and absorbing homomorphisms. Then we use this result to verify the conjecture when B is a countable direct sum of matrix algebras and the embeddings of B into A ₁ and A ₂ are quasiunital. Inspired by the proof we achieve the following nice classification result: A separable C *‐algebra B is a countable direct sum of matrix algebras if and only if the unitary group of the multiplier algebra U M (B) is compact in the strict topology. Finally we prove the conjecture when the amalgamated free product has the property that any asymptotically split extension of A _{1 ?? B} A ₂ is split. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A criterion for the <Emphasis Type="Italic">F</Emphasis> <Subscript> <Emphasis Type="Italic">π</Emphasis> </Subscript>-residuality of free products with amalgamated cyclic subgroup of nilpotent groups of finite ranks

Let G be the free product of nilpotent groups A and B of finite rank with amalgamated cyclic subgroup H, H ≠ A and H ≠ B. Suppose that, for some set π of primes, the groups A and B are residually F_π, where F_π is the class of all finite p-groups. We prove that G is residually F_π if and only if H is F_π-separable in A and B.     

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.     

Generalized involution models for wreath products

We prove that if a finite group H has a generalized involution model, as defined by Bump and Ginzburg, then the wreath product H ? S _n also has a generalized involution model. This extends the work of Baddeley concerning involution models for wreath products. As an application, we construct a Gel’fand model for wreath products of the form A ? S _n with A abelian, and give an alternate proof of a recent result due to Adin, Postnikov and Roichman describing a particularly elegant Gel’fand model for the wreath product ? _r ? S _n . We conclude by discussing some notable properties of this representation and its decomposition into irreducible constituents, proving a conjecture of Adin, Postnikov and Roichman.     

Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture

Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH ₀(A) and HH ₀(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.     

On the Howson Property of HNN-Extensions with Nilpotent Base Group and Amalgamated Free Products of Nilpotent Groups

We give necessary and sufficient conditions under which an amalgamated free product of finitely generated nilpotent groups is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated). Also we prove that if G = ? t, K | t ^?1 At = B ?, where K is a finitely generated and infinite nilpotent group and A, B non-trivial infinite proper subgroups of K, then G is not a Howson group. The problem of deciding when an ascending HNN-extension of a finitely generated nilpotent group is a Howson group is still open.     

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Free products of kω-topological groups with normal amalgamation

It is shown in this paper that if A is a closed normal subgroup of k_ω-topological groups G and H, then the free product of G and H with A amalgamated, G1_AH, exists, is Hausdorff and indeed a k_ω-group.     

Free product actions with relative property (T) and trivial outer automorphism groups

We show that every non-amenable free product of groups admits free ergodic probability measure preserving actions which have relative property (T) in the sense of S. Popa (2006) [Pop06, Def. 4.1]. There are continuum many such actions up to orbit equivalence and von Neumann equivalence, and they may be chosen to be conjugate to any prescribed action when restricted to the free factors. We exhibit also, for every non-amenable free product of groups, free ergodic probability measure preserving actions whose associated equivalence relation has trivial outer automorphisms group. This gives, in particular, the first examples of such actions for the free group on 2 generators.     

Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions

We introduce analogs of the Hopf algebra of Free quasi-symmetric functions with bases labeled by colored permutations. When the color set is a semigroup, an internal product can be introduced. This leads to the construction of generalized descent algebras associated with wreath products Γ?S_n and to the corresponding generalizations of quasi-symmetric functions. The associated Hopf algebras appear as natural analogs of McMahon’s multisymmetric functions. As a consequence, we obtain an internal product on ordinary multi-symmetric functions. We extend these constructions to Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type B.