Growth rate of an endomorphism of a group

In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ 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

The Tarksi Theorems, Extensions to Group Rings and Logical Rigidity (Logic, Algebraic system, Language and Related Areas in Computer Science)

This is from a talk presented at the Kobe Conference 2022 held in Kobe, Japan.The famous Tarski theorems state that all free groups heve the same elementary theory. In 2019 I gave a talk at the Kobe conference explaining the Tarski theorems and the accompanying language. Subsequently in [FGRS 1, 2, 3] and [FGKRS] the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R was examined. These are relative to an appropriate logical language L₀, L₁, L₂ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS 1]. In [FGRS 1] it was proved that if R[G] is elementarily equivalent to S[H] with respect to L₂, then simultaneously the group G is elementarily equivalent to the group H with respect to Lo, and the ring R is elementarily equivalent to the ring S with respect to L₁. We then let F be a rank 2 free group and Z be the ring of integers. Examining the universal theory of the free group ring Z[F] the hazy conjecture was proved that the universal sentences true in Z[F] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in Z[F] modified appropriately for group theory. Finally we mention logical group rigidity. A group G is logically rigid if being elementary equivalent to G is equivalent to being isomorphic to G. In this paper we survey all of these findings

Constructing Geometries for Group Control: Methods for Reasoning about Social Behaviors

Social behaviors in groups has been the subjects of hundreds of studies in a variety of research disciplines, including biology, physics, and robotics. In particular, flocking behaviors (commonly exhibited by birds and fish) are widely considered archetypical social behavioris and are due, in part, to the local interactions among the individuals and the environment. Despite a large number of investigations and a significant fraction of these providing algorithmic descriptions of flocking models, incompleteness and imprecision are readily identifiable in these algorithms, algorithmic input, and validation of the models. This has led to a limited understanding of the group level behaviors. Through two case-studies and a detailed meta-study of the literature, this dissertation shows that study of the individual behaviors are not adequate for understanding the behaviors displayed by the group. To highlight the limitations in only studying the individuals, this dissertation introduces a set of tools, that together, unify many of the existing microscopic approaches. A meta-study of the literature using these tools reveal that there are many small differences and ambiguities in the flocking scenarios being studied by different researchers and domains; unfortunately, these differences are of considerable significance. To address this issue, this dissertation exploits the predictable nature of the group’s behaviors in order to control the given group and thus hope to gain a fuller understanding of the collective. From the current literature, it is clear the environment is an important determinant in the resulting collective behaviors. This dissertation presents a method for reasoning about the effects the geometry of an environment has on individuals that exhibit collective behaviors in order to control them. This work formalizes the problem of controlling such groups by means of changing the environment in which the group operates and shows this problem to be PSPACE-Hard. A general methodology and basic framework is presented to address this problem. The proposed approach is general in that it is agnostic to the individual’s behaviors and geometric representations of the environment; allowing for a large variety in groups, desired behaviors, and environmental constraints to be considered. The results from both the simulations and over 80 robot trials show (1) the solution can automatically generate environments for reliably controlling various groups and (2) the solution can apply to other application domains; such as multi-agent formation planning for shepherding and piloting applications

Amalgam decompositions for one-relator groups

AbstractIt is known that a one-relator group G with at least three generators admits a proper free product with amalgamation decomposition (A★B;C) with finitely generated factors. We call such a decomposition a Baumslag–Shalen decomposition. Little is known of the exact nature of the factors. Here we first prove that if the amalgamated subgroup is finitely presented, in particular free, then each factor is finitely presented. Using this we show that if G is a torsion-free one-relator group with Baumslag–Shalen decomposition (A★B;C) with C free, then each factor is homologically equivalent to either a free group or a one-relator group. Furthermore, sufficient conditions are obtained for G to be cyclically pinched if the factors are free groups

Surface Groups Within Baumslag Doubles

A conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V is an element of F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X(2)Y(2) for some X, V is an element of F. G can contain no non-orientable surface group of smaller genus