We consider (graph-)group-valued random element ξ, discuss the properties of a mean-set 𝔼(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.

is finite, and a uniform X-lattice if Γ\X is a finite graph, non-uniform otherwise. In [C2] and [BCR] we gave the necessary and sufficient conditions for the existence of X-lattices. In this work we determine how to construct pairs of tree lattice subgroups. In the setting of topological covering theory, there is a correspondence between coverings, p : X → A, and subgroups of π1(A), and this gives essentially ‘one’ subgroup up to conjugacy. Here with lattices given by their quotient graphs of groups, we are in an ‘orbifold’ setting where coverings have an extra ingredient, namely isotropy groups. We make use of this additional data to construct lattice subgroups by using subgroups of isotropy (vertex) groups. This will give rise to zero, one, finitely many, or infinitely many possible subgroups up to isomorphism. In fact in [CR1] the authors exhibited infinite ascending chain of lattice subgroups, all with the same quotient graph. We describe several methods (sections 3-5) for constructing a pair of Xlattices (Γ′,Γ) with Γ ≤ Γ′, starting from ‘edge-indexed graphs’ (A′, i′) and (A, i) which will correspond to the edge-indexed quotient graphs of their (common) universal covering tree by Γ′ and Γ respectively. Our techniques are a combination of topological graph theory, covering theory for graphs of groups ([B]), and covering theory for edge-indexed graphs developed in [C1] and [BCR]. As an application, we show (section 5) that a nonuniform X-lattice Γ contains an infinite chain of subgroups Λ1 < Λ2 < Λ3 < . . . where each Λk is a uniform Xj-lattice, Xk a subtree of X. Let φ : (A, i) → (A′, i′) be a covering of edge-indexed graphs (defined in section 2). We wish to determine if it is possible to extend φ to a covering

In this paper, we apply quantum algorithms to solve problems concerning fixed points and invariant subgroups of automorphisms. These efficient algorithms invoke a quantum algorithm which computes the intersection of multiple unsorted multisets whose elements originate from the same set. This intersection algorithm is an application of the Grover search procedure.

In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map f : M 7→ M on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient.We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent.

In this paper, we study two digital signature algorithms, the DSA and ECDSA, which have become NIST standard and have been widely used in almost all commercial applications. We will show that the two algorithms are actually ‘the same’ algebraically and propose a generic algorithm such that both DSA and ECDSA are instances of it. By looking at this special angle through the generic algorithm, we gain a new insight into the two algorithms DSA and ECDSA. Our new proposed digital signature algorithm is described generically using a group G and a map to Number : G → ℤ. As an illustration, we choose G to be a group of non-singular circulant matrices over finite field and describe a totally new concrete digital signature algorithm.

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them.

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Hyperbolic groups have been studied in various fields in mathematics. They appear in contexts as diverse as geometric group theory, function theory (as Fuchsian groups) and algebraic topology (as fundamental groups of compact hyperbolic surfaces). Hyperbolic groups possess geometrical properties well suited for the study of homological finiteness conditions. In this paper we will prove some of these results via free resolutions obtained from the Rips-complex.

Recall that asymptotic density is a method to compute densities and/or probabilities within infinite finitely generated groups. If is a group property, the asymptotic density determines the measure of the set of elements which satisfy . Is this asymptotic density equal to 1, we say that the property is generic in G . is called an asymptotic visible property, if the corresponding asymptotic density is strictly between 0 and 1. If the asymptotic density is 0, then is called negligible . We call a group property suitable if it is preserved under isomorphisms and its asymptotic density exists and is independent of finite generating systems. In this paper we prove that there is an interesting connection between the strong generic free group property of a group G and its subgroups of finite index.