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The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

Through the glasses of didactic reduction: We consider a (periodic) tessellation $\Delta$ of either Euclidean or hyperbolic $n$-space $M$. By a piecewise isometric rearrangement of $\Delta$ we mean the process of cutting $M$ along corank-1 tile-faces into finitely many convex polyhedral pieces, and rearranging the pieces to a new tight covering of the tessellation $\Delta$. Such a rearrangement defines a permutation of the (centers of the) tiles of $\Delta$, and we are interested in the group $PI(\Delta)$ of all piecewise isometric rearrangements of $\Delta$. In this paper we offer: a) An illustration of piecewise isometric rearrangements in the visually attractive hyperbolic plane, b) an explanation how this is related to Richard Thompson's groups, c) a chapter on the structure of the group pei$(\mathbb Z^n)$ of all piecewise Euclidean rearrangements of the standard tessellation of $\mathbb R^n$ by unit-cubes, and d) results on the finiteness properties of some subgroups of pei$(\mathbb Z^n)$., Comment: arXiv admin note: substantial text overlap with arXiv:1606.07728

The observation that the 0-dimensional Geometric Invariant $\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\'e's limit set $\Lambda (\Gamma)$ of a discrete group $\Gamma $ of M\"obius transformations (which contains the horospherical limit set of $\Gamma $) to the roots of tropical geometry (closely related to $\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm., Comment: This is the final published version

Thompson's group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute † m .F / and † m .F I Z/, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show that † m .F / D † m .F I Z/ .A s an application, we show that, for every m, F has subgroups of type Fm� 1 which are not of type FPm (thus certainly not of type Fm).

The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invariants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schuetz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova to derive new information about subgroups of Thompson's group F, namely that F has subgroups F_m which are not of type F_{m+1}., 9 pages

It is known that the geometric invariants of a group G (which contain information on finiteness properties of certain submonoids and normal subgroups of G ) have a description in terms of the vanishing of group homology of G with Novikov-ring-coefficients [see J.-C. Sikorav, Homologie de Novikov associee a une classe de cohomologie reelle de degre un, These Orsay, 1987; R. Bieri, The geometric invariants of a group, in: G.A. Niblo, M.A. Roller (Eds.), Geometric Group Theory, in: London Math. Soc. Lecture Notes Series, vol. 181, Cambridge University Press, Cambridge, 1993; R. Bieri, R. Strebel, Geometric invariants for discrete groups, manuscript-preprint of a monograph (in preparation)], and [R. Bieri, R. Geoghegan, Kernels of actions on non-positively curved spaces, in: P.H. Kropholler, G. Niblo, R. Stohr (Eds.), Geometry and Cohomology in Group Theory, in: London Math. Soc. Lecture Notes Series, vol. 252, Cambridge University Press, Cambridge, 1998, pp. 24–38]. In a recent paper Kochloukova [D. Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups (submitted for publication)] uses this to prove a conjecture of E. Rapaport-Strasser on knot-like groups. We extend her approach to establish a rather general relationship between deficiency and the geometric invariants of a group.

We describe new topological invariants of the classical action of SL 2(ℤ[1/m]) on the hyperbolic plane, where m is a positive integer. One defines the notion of this action (or any action of a discrete group on a CAT(0) metric space) as being 'controlled s-connected' where s is an integer ≥−1. In the case of this classical action, we show it is (s − 2)-connected but not (s − 1)-connected where s is the number of different primes dividing m. It turns out that the behavior of this action is quite different over rational and irrational endpoints, with the bounds on connectivity occurring only at the rational end points. The very lowest case, m=1, reduces to elementary and seemingly innocuous statements about how SL 2(ℤ) acts on the hyperbolic plane. Extensions involving other Fuchsian groups are also given.

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group., 32 pages, no figure. 1->2 Major changes (the 13-page first version, authored by Y.C. and L.G., was entitled "On infinitely presented soluble groups") 2->3 some changes including cuts in Section 4

1.1 Background. Throughout this note Q stands for a finitely generated multiplicative Abelian group of torsion-free rank n, R for a commutative ring with 1, and M for a finitely generated RQ-module. The geometric invariant ΣM of M was introduced in [BS 80/81]. It can be viewed as a subset of the R-vector space of all (additive) charakters of Q, Q = Hom(Q,R) ∼= R, as follows: For every character χ : Q → R one considers the submonoid Qχ = {q ∈ Q | χ(q) ≥ 0} of Q and puts ΣM : = {χ | M is finitely generated over RQχ}. Note that 0 ∈ ΣM . It is often convenient to work with the complement ΣM of ΣM in Q . The geometric invariant ΣM has been investigated for two reasons. Firstly, if R is a Dedekind domain, then ΣM turns out to be a polyhedral 1 subset of Q. This rather subtle fact was conjectured, for R a field, by G.M. Bergman [Be 71] and established by John R.J. Groves and the first author in [BG 84]; it opens the possibility for computations and imposes arithmetic restrictions on automorphisms of M . Secondly, for R = Z,ΣM contains interesting information on the (metabelian) groups G which are extensions of M by Q. In [BS 80] it is proved that G has a finite presentation if and only if ΣM ∪ −ΣM = Q. A number of attempts have been undertaken to extend this result to a charakterization of the higher dimensional finiteness property

The following six new genera of Sagittidae, with type species listed in parentheses, are proposed: Adhesisagitta (Sagitta hispidu Conant, 1985), Demisagitta (Sagitta demipenna Tokioka and Pathansali, 1963), Decipisugitta (Sugittu decipiens Fowler, 1905). Tenuisugittu (Sagitta tenuis Conant, 1986), Abacisagitta (Sagitta pulchra Doncaster, 1903), Oculosagitta (Sagitta megalophthalma Dallot and Ducret, 1969).

We attach to every finitely generated nilpotent-by-Abelian-by-finite group G a closed subset σ(G) of a sphere and demonstrate that this geometric invariant measures certain finiteness properties of G, such as being finitely presented, constructible, coherent, or polycyclic-by-finite.

1.1 Finitely presented soluble groups have been investigated by several authors. Roughly speaking, their results deal with two aspects: with the subgroup structure of finitely presented soluble groups [2], [4], [7], [18], and with soluble varieties whose non-cyclic relatively free group are infinitely related [16], [3]. Here we attack the problem of recognizing which finitely generated soluble groups are finitely related in a somewhat more systematic way: We show that all finitely presented soluble groups have a certain structural property which is inherited by homomorphic images (whether this holds for the property of being finitely presented itself is an old problem of P. Hall's and is still open). 1.2 The methods of [2] and [4] made it clear that even in the soluble case the HNN-construction is an important tool for obtaining finitely presented groups. Recall that every group B containing a pair of isomorphic subgroups 0 : S---~ T is embedded in the HNN-group

In an earlier paper [ 11, the authors investigated, as a major component of their final result, a special case of the following question. Let R be a commutative ring with unity, G a finitely generated abelian group and A4 a finitely generated RG-module. Consider the tensor power @“, M as an RG-module via the diagonal action of G. Under what conditions is @“, M a finitely generated RG-module? The answer obtained in [ 1 ] was for the case when R is a field and used an invariant Z;M introduced by the first author and Strebel in [3]. We briefly describe CL. Firstly, if n is the torsion-free rank of G, we can identify Hom(G, [w) with [w”. Defining two elements of Hom(G, Iw) to be equivalent if they differ by a positive real multiple, we obtain a natural projection of Hom(G, [w)\(O) onto the sphere s” I. Denote the equivalence class of XE Hom(G, 1w) by [x] and define GcXI = G,= { ge G: x(g) 30). Then Z:, is the subset of S”’ consisting of all those [x] for which M is a finitely generated RG,-module and CL is the complement of C, in S”‘. We say that M is k-tame if there do not exist [x, I,..., [xk ] in Zh such that x1 + “. + xk = 0. Then the partial answer referred to above (Theorem 3.4 of [l]) is

The teeth of Flaccisagitta hexaptera are capable of penetrating the exoskeleton of copepods. The grasping spines, anterior, and posterior teeth are all capable of piercing the epidermis of larval fish as seen in a series of scanning electron micrographs. A new type of anterior tooth structure is observed. A chemosensory function is suggested for a new set of pores found between the sets of anterior teeth. The secretory nature of the vestibular pit is clearly seen for the first time.

The geographic extents of 27 species of Chaetognatha in the Pacific have been inferred from analyses of 2,900 plankton samples taken at more than 900 stations. Two species of chaetognaths—Sagitta elegans and S. maxima—are found in the Pacific only in the Subarctic Water Mass. Four species—Sagitta enflata, S. hexaptera, S. pacifica, and Pterosagitta draco—occur in the Pacific Equatorial and Central Water Masses. Seven species—Sagitta robusta, S. regularis, S. ferox, Krohnitta pacifica, Sagitta bedoti, S. neglecta, and S. pulchra—occur in the Equatorial and Western Central Water Masses. The latter three species appear to be disjunct equatorial species that are absent from the central equatorial Pacific. Sagitta pseudoserratodentata occupies the North Pacific Central Water Mass. S. californica, essentially a warm-water cosmopolite, is very rare in the Equatorial Water Mass. The Peru and California Currents are inhabited by Sagitta friderici. A close relative, S. crassa, lives in the coastal waters of Japan. Sagitta minima, Sagitta species (serratodentata group), Sagitta decipiens, S. lyra, and Krohnitta subtilis, species which have their maximum abundance at about 200 meters (somewhat deeper than all of the preceding), appear to be limited in the north at about 45° N. Lat. Sagitta planctonis, S. macrocephala, and Eukrohnia fowleri, living primarily at depths of 500 meters and greater, have been taken in the present study as far north as 55° N. Lat., both in the Bering Sea and Gulf of Alaska. The very deep living species, Heterokrohnia mirabilis, was taken at a single station in the Eastern Equatorial Pacific. The sharpest chaetognath boundary yet found in the Pacific occurs in the Eastern Pacific at 18° to 20° N. Lat. and is associated with pronounced temperature, oxygen, and probably salinity gradients. Species distributions analogous to those of the Chaetognatha are found in the euphausiids, copepods, and dinoflagellates among other groups of organisms.

The interaction between the members of the marine biosphere and the various chemical species in sea water commands attention as one of the focal points in studies of productivity and sedimentation. For extending our understanding of biogeochemical processes, the permutations upon the composition of sea water produced by the detonation of nucIear devices offer great promise. First, the introduction of minute amounts of easily measurable radioactive substances provides ready access to the paths of such substances from the hydrosphere to the organisms and subsequently from organism to organism. Single elements, as well as suites of elements, can readily be studied by well-established radiochemical techniques for their isolation and quantitative analysis. Second, time studies on the retention of assimilated substances can be undertaken. The biological half lives of elements in marine organisms, in conjunction with the respective concentration factors, are of paramount importance in considerations upon the long-time cffccts of nuclear tests. The purpose of our field studies was to make a preliminary survey of the distribution of fission products between the marinc waters and members of biosphere. The work was limited in two ways : 1) The time available for the sampling of both water and organisms was secondary to the main objectives of the Scripps Institution of Oceanography investigations, and stations could not be made in every definitive area of interest; 2) owing to the lack of personnel, equipment, and space aboard the vessels, chemical separations of the fission products were not made. This latter limitation prevented studies on the more intimate behavior of the various nuclear products from the explosion.