We review some known results and open problems related to the growth of groups. For a finitely generated group Γ, given whenever necessary together with a finite generating set, we discuss the notions of (A) uniformly exponential growth, (B) growth tightness, (C) regularity of growth series, (D) classical growth series (with respect to word lengths), (E) growth series with respect to weights, (F) complete growth series, (G) spectral radius of simple random walks on Cayley graphs. From the modern point of view a dynamical system is a pair (G, X) where G is a group (or a semi-group) and X is a set on which G acts; introducing different structures on G and X we get different directions for the Theory of Dynamical Systems. Many of the dynamical properties of the pair (G, X) depend on appropriate properties of the group G. In this paper we discuss such notions for a group G as growth, entropy, amenability

Abstract. The quotient of a biautomatic group by a subgroup of the center is shown to be biautomatic. The main tool used is the Neumann-Shapiro triangulation of S n−1, associated to a biautomatic structure on Z n. As an application, direct factors of biautomatic groups are shown to be biautomatic. Biautomatic groups form a wide class of finitely presented groups with interesting geometric and computational properties. These groups include all word hyperbolic groups, all fundamental groups of finite volume Euclidean and hyperbolic orbifolds, and all braid groups [E...]. A biautomatic group satisfies a quadratic isoperimetric inequality, has a word problem solvable in quadratic time, and has a solvable conjugacy problem. The class of biautomatic groups has several interesting closure properties. For instance, the centralizer of a finite subset of a biautomatic group is biautomatic, as is the center of the whole group [GS]. Also, biautomatic groups are closed under direct products [E...]. The theory of biautomatic groups is briefly reviewed below. We present a technique for putting biautomatic structures on central quotients of biautomatic groups: Theorem A. Let G be a biautomatic group, and let C be a subgroup of ZG, the center. Then G/C is biautomatic. This result has several applications. Our first application answers a question posed by Gersten and Short [GS, cf. proposition 4.7]: Theorem B. Direct factors of biautomatic groups are biautomatic. Proof. Suppose G × H is biautomatic. The centralizer of H is G × ZH, and this is a biautomatic group by [GS, corollary 4.4]. Then ZH is a subgroup of the center of G×ZH, so by theorem A, G × ZH/ZH = G is biautomatic. Several recent discoveries have pointed to the useful concept of poison subgroups. For instance, the group Z 2 is poison to word hyperbolic groups: if a group contains a Z 2 subgroup it cannot be word hyperbolic. A wider class of subgroups poison to word hyperbolicity are those which have an infinite index central Z subgroup [CDP, corollaire 7.2]. Our next theorem says that for biautomatic groups, this class of poison subgroups completely collapses to Z 2:

Abstract: A finite complete rewriting system for a group is a finite presentation which gives an algorithmic solution to the word problem. Finite complete rewriting systems have proven to be useful objects in geometric group theory, yet little is known about the geometry of groups admitting such rewriting systems. We show that a group G with a finite complete rewriting system admits a tame 1-combing; it follows (by work of Mihalik and Tschantz) that if G is an infinite fundamental group of a closed irreducible 3-manifold M, then the universal cover of M is R 3. We also establish that a group admitting a geodesic rewriting system is almost convex in the sense of Cannon, and that almost convex groups are tame

Abstract. The logic L(Qu) extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying... belongs to the set C”). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [5]. It is shown that, as in the case of automatic structures [19], also modulocounting quantifiers as well as infinite cardinality quantifiers (“there are κ many elements satisfying...”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of L(Qu) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold. §1. Introduction. Automatic structures were introduced in [13, 16]. The idea goes back to the concept of automatic groups [9]. Roughly speaking, a structure is called automatic if the elements of the universe can be represented as words from a regular language and every relation of the structure can be recognized by a finite state automaton with several heads that proceed synchronously.

. Most imperative languages only offer arrays as "first-class" data structures. Other data structures, especially recursive data structures such as trees, have to be manipulated using explicit control of memory, i.e., through pointers to explicitly allocated portions of memory. We believe that this severe limitation is mainly due to historical reasons, and this paper will try and demonstrate that modern analysis techniques, such as data flow analysis, allow to cope with the compilation problems associated with recursive data structures. As a matter of fact, recursion in the flow of control also is a current open issue in automatic parallelization: to our knowledge, no theory allows the parallelization of, e.g., recursive Pascal programs. This paper uniformly handles both issues. We propose a kernel language that manipulates recursive data structures in an elegant, algebraic way. In this preliminary work, both data- and control- recursive structures are restricted, so that a data flow a...

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A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This article surveys results for combable groups, in particular in the case where the combing is a formal language.

Abstract. The purpose of this paper is to investigate torsionfree groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats and relatively thin triangles, as defined by Hruska [17]. Our approach is to seek results analogous to those of Sela, Kharlampovich and Miasnikov for free groups and to those of Sela (and Rips and Sela) for torsion-free hyperbolic groups. This paper is the first in a series. In this paper we extract an R-tree from an asymptotic cone of certain CAT(0) spaces. This is analogous to a construction of Paulin, and allows a great deal of algebraic information to be inferred, most of which is left to future work. 1.

Substitution tilings have been discussed now for at least twenty-five years, initially motivated by the construction of hierarchical non-periodic structures in the Euclidean plane [?,?,?,?]. Aperiodic sets of tiles were often created by forcing these structures to emerge. Recently, this line was more or less

In the last several years a remarkable interplay between geometry, group theory, and the theory of formal languages has led to developments including the introduction of automatic groups [Ep+], hyperbolic groups [Gro], and geometric and language-theoretic characterizations of finitely generated virtually free groups