Abstract:
This paper is about geometric and topological properties of a proper CAT(0) space $X$ which is cocompact - i.e. which has a compact generating domain with respect to the full isometry group. It is shown that geodesic segments in $X$ can "almost" be extended to geodesic rays. A basic ingredient of the proof of this geometric statement is the topological theorem that there is a top dimension $d$ in which the compactly supported integral cohomology of $X$ is non-zero. It is also proved that the boundary-at-infinity of $X$ (with the cone topology) has Lebesgue covering dimension $d-1$. It is not assumed that there is any cocompact discrete subgroup of the isometry group of $X$; however, a corollary for that case is that "the dimension of the boundary" is a quasi- isometry invariant of CAT(0) groups. (By contrast, it is known that the topological type of the boundary is not unique for a CAT(0) group.)

Abstract:
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\"{o}bius 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.

Abstract:
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}.

Abstract:
Given a set S equipped with a binary operation (we call this a "bracket algebra") one may ask to what extent the binary operation satisfies some of the consequences of the associative law even when it is not actually associative? We define a subgroup Assoc(S) of Thompson's Group F for each bracket algebra S, and we interpret the size of Assoc(S) as determining the amount of associativity in S - the larger Assoc(S) is, the more associativity holds in S. When S is actually associative, Assoc(S) = F; that is the trivial case. In general, it turns out that only certain subgroups of F can occur as Assoc(S) for some S, and we describe those subgroups precisely. We then explain what happens in some familiar examples: Lie algebras with the Lie bracket as binary operation, groups with the commutator bracket as binary operation, the Cayley numbers with their usual multiplication, as well as some less familiar examples. In the case of a group G, with the commutator bracket as binary operation, it is better to think of the "virtual size of G", determined by all the groups Assoc(H) such that H is a subgroup of finite index in G. This gives a way of partitioning groups into "small", "intermediate" and "large" - a partition suggestive of, but different from, traditional measures of a group's size such as growth, isoperimetric inequality and "amenable vs. non-amenable"

Abstract:
Using a theorem of L\"uck-Reich-Rognes-Varisco, we show that the Whitehead group of Thompson's group T is infinitely generated, even when tensored with the rationals. To this end we describe the structure of the centralizers and normalizers of the finite cyclic subgroups of T, via a direct geometric approach based on rotation numbers. This also leads to an explicit computation of the source of the Farrell-Jones assembly map for the rationalized higher algebraic K-theory of the integral group ring of T.

Abstract:
This is the second of two papers but has been written so as to have minimal dependence on the first paper (which is also on this archive). Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Assume G is of type F_n (type F_1 is finitely generated, type F_2 is finitely presented etc.) The "boundary", bdM, of M at infinity has two customary topologies - the compact "cone" topology and the Tits metric topology. We associate with any isometric action of G on M two subsets of bdM, both dependent on n. These subsets consist of those points of bdM at which - in two senses - the action is "controlled (n-1)-connected". One of these sets is open in the Tits metric topology. Even in classical cases like familiar groups of isometries of the hyperbolic plane or of a locally finite tree these sets seem to be new and interesting invariants. The "SIGMA-theory" of Bieri-Neumann-Strebel-Renz is recovered in the special case in which M is G(abelianized) tensor R with the translation action.

Abstract:
We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for the Euler class (integer coefficients) of the normal bundle to a smooth circle action without fixed points on a manifold. In the special case of a 3-dimensional Seifert fibered space, this formula is particularly effective. \~

Abstract:
Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The cocompact actions form an open subset of R. 2. The cocompact actions with discrete orbits whose point-stabilizers have type F_n form an open subset of the subspace of R consisting of all actions with discrete orbits. (F_1 means finitely generated, F_2 means finitely presented etc.) The key idea is to introduce a new "controlled topology" invariant of such actions - dependent on n - which is unfamiliar when the orbits are not discrete but which becomes familiar (cf 2.) when the orbits are discrete. (This is the first of two papers.)

Abstract:
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-manifold case, Wright's method showed that when a one-ended, simply connected, locally compact ANR X with pro-monomorphic fundamental group at infinity admits an action of Z by covering transformations then the fundamental group at infinity of X is (up to pro-isomorphism) an inverse sequence of finitely generated free groups. We improve upon this latter result, by showing that X must have a stable finitely generated free fundamental group at infinity. Simple examples show that a free group of any finite rank is possible. We also prove that if X (as above), admits a non-cocompact action of Z+Z by covering transformations, then X is simply connected at infinity. Corollary: Every finitely presented one-ended group G which contains an element of infinite order satisfies exactly one of the following: 1) G is simply connected at infinity; 2) G is virtually a surface group; 3) The fundamental group at infinity of G is not pro-monomorphic. Our methods also provide a quick new proof of Wright's open manifold theorem.

Abstract:
In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CW- complexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik- Schnirelmann category of a closed 1-form and to the existence of a non- singular closed 1-form in a given cohomology class on a high-dimensional closed manifold.