On measure homology of mildly wild spaces

It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of \({\mathbb{R}^4}\). According to the theorem of Shelah [10] such space can not be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X G for every countable group G.

The constructibility axiom, V = L, is a statement in mathematics that is independent of the ZFC axioms of set theory. One of its consequences is (μ) = measurable cardinals do not exist. We assume (μ). Let R be a ring, and let G be a right R-module. We say that G is slender if for each indexed set {R i | i ∈ ℐ} (of nonmeasurable cardinality) such that R ≅ R i for each i ∈ ℐ, the canonical map is an isomorphism. Reduced countable torsion-free abelian groups are slender groups. Nonzero slender injective right R-modules do not exist. We say that G is self-slender if for each (nonmeasurable) cardinal ℵ, the canonical map is an isomorphism. Reduced countable torsion-free abelian groups G are self-slender. We show in this article that if we assume (μ), then for each self-slender right R-module G there is a category G-coPlex in which G is a slender injective object and such that End(G)-Mod is dual to G-coPlex. Thus there are many nonzero slender injective objects.

In this paper we give a number of explicit constructions for II 1 _1 factors and II 1 _1 equivalence relations that have prescribed fundamental group and outer automorphism group. We construct factors and relations that have uncountable fundamental group different from R + ∗ \mathbb {R}_{+}^{\ast } . In fact, given any II 1 _1 equivalence relation, we construct a II 1 _1 factor with the same fundamental group. Given any locally compact unimodular second countable group G G , our construction gives a II 1 _1 equivalence relation R \mathcal {R} whose outer automorphism group is G G . The same construction does not give a II 1 _1 factor with G G as outer automorphism group, but when G G is a compact group or if G = S L n ± R = { g ∈ G L n R ∣ det ( g ) = ± 1 } G=\mathrm {SL}^{\pm }_n\mathbb {R}=\{g\in \mathrm {GL}_n\mathbb {R}\mid \det (g)=\pm 1\} , then we still find a type II 1 _1 factor whose outer automorphism group is G G .

As Avez showed (in 1970), the fundamental group of a compact Riemannian manifold of nonpositive sectional curvature has exponential growth if and only if it is not flat. After several generalizations from Gromov, Zimmer, Anderson, Burger and Shroeder, the following theorem was proved by Adams and Ballmann (in 1998). Theorem Let X be a proper CAT(0) space. If Γ is an amenable group of isometries of X, then at least one of the following two assertions holds: Γ fixes a point in ∂X (boundary of X). X contains a Γ-invariant flat (isometric copy of Rn, n ≥ 0). Following an idea of my PhD advisor Nicolas Monod, I tried to generalize this theorem in the context of goupoids, in this case Borel G-spaces and countable Borel equivalence relations. This lead me to study the notion of Borel fields of metric spaces, which turns out to be a suitable context to define an action of a countable Borel equivalence relation. A field of metric spaces over a set Ω is a family {(Xω,dω)} ω∈Ω of nonempty metric spaces denoted by (Ω,X•). We introduced as S( Ω,X•) the set of maps Such maps are called sections. If Ω is a Borel space, we can define a Borel structure on a field of metric spaces to be a subset Lℒ( Ω,X•) of S( Ω,X•) satisfying these three conditions For all f, g ∈ ℒ(Ω,X•), the function Ω → R, ω → dω(f(ω), g(ω)) is Borel. If h ∈ S(Ω,X•) is such that the function Ω → R, ω → dω(f(ω), h(ω)) is Borel for all f ∈ ℒ(Ω,X•), then h ∈ ℒ(Ω,X•). There exists a countable family of sections {fn}n≥1 ⊆ ℒ(Ω,X•) such that {fn (ω)}n≥1 = Xω for all ω ∈ Ω. This definition is consistent with more classical definitions of Borel fields of Banach spaces or of Borel fields of Hilbert spaces. The notion of a Borel field of metric spaces has been used in convex analysis and in economy. As said before, we can define an action of a countable Borel equivalence relation ℛ ⊆ Ω2 on a Borel field of metric spaces (Ω,X•) in a natural way. It's determined by a family of bijectives maps {α(ω, ω') : Xω → Xω'}(ω,ω')∈ℛ such that For all (ω,ω'), (ω',ω") ∈ ℛ the following equality is satisfied α(ω', ω") ◦ α(ω, ω') = α(ω, ω"). For all f, g ∈ ℒ(Ω,X), the function ℛ → R, (ω, ω') → dω(f(ω), α(ω', ω)g(ω')) is Borel. Zimmer (1977) introduced the notion of amenability for ergodic G-spaces and equivalence relations, of which we obtained the first generalization (in collaboration with Philippe Henry). Theorem Let R be a countable, Borel, preserving the class of the measure, ergodic and amenable equivalence relation on the probability space Ω acting on a Borel field ( Ω,X•) of proper CAT(0) spaces with finite topological dimension. Then at least one of the following assertions is true: There exists an ℛ-invariant Borel section ξ ∈ L(Ω,∂X•). There exists an ℛ-invariant Borel subfield (Ω, F•) of (Ω,X•) consisting of flat subsets. And the second generalization for amenable ergodic G-spaces. Theorem Let G be a locally compact second countable group, Ω a preserving class of the measure, ergodic amenable G-space, X a proper CAT(0) space with finite topological dimension and α : G × Ω → Iso(X) a Borel cocycle. Then at least one of the following assertions is true: There exists an α-invariant Borel function ξ : Ω → ∂X. There exists an α-invariant borelian subfield (Ω, F•) of the trivial field (Ω, X) consisting of flat subsets. If we consider (Ω,μ) to be a strong boundary of the group G, the cocycle α to come from an action of G on X, and X to have flats of at most dimension 2, then we can conclude the following. Theorem Let G be a locally compact second countable group, (Ω,μ) a strong boundary of G, X a proper CAT(0) space with finite topological dimension and whose flats are of dimension at most 2. Let suppose that G acts by isometry on X. Then at least one of the following assertions is true: There exists a G-equivariant Borel function ξ: Ω → ∂X. There exists a G-invariant flat F in X. The proof of the three theorems are strongly based on properties of Borel field of metric spaces that we prove in this thesis.

Let G be a group with presentation such that each generator occurs at most countably many times in the set of relations. Then for all n 3. G is ,I-ic. In particular, for all n 2 3, countable groups have n-ic presentations. Neuwirth (I) calls a group n-ic if it admits a presentation in which each generator appears exactly n times in the set of relations; such a presentation is also called n-ic. He goes on to show that if G is the fundamental group of a closed n-manifold then G * F is n-ic for some free F. Hoare (unpublished; see (2, p. 1461) has shown for n = 3 that G itself is 3-ic (cubic), and Montesinos (3) extends thls result to 3-manifolds with boundary. Here we show that any group that admits a presentation in which each generator occurs at most countably many times in the set of relations is n-ic.

A method is shown to construct ANR-topologies on topological groups of suitable homotopy type. It is well known that every simple countable CW-complex with vanishing Postnikov invariants has the homotopy type of a topological Abelian group G, that can be taken as realization of a countable simplicial Abelian group and hence is a countable CW-complex [5, Ch.V]. In particular this means that the homotopy groups 7rn(G) can be chosen as an arbitrarily prescribed sequence of countable Abelian groups. However, there are situations where it would be preferable to have G as an ANR-space instead of a CW-complex. One would guess that a suitable topology exists on G, because by subdivision it can be turned into a polyhedron and then can get the metric topology, but the subdivision process necessarily leaves the range of simplicial Abelian groups and need not produce a group topology. Instead, we follow a method developed by Cauty in [1]. Theorem. Let G be a locally contractible, not necessarily Abelian group, every open subset of which is a-compact, such that G is the union of a sequence of finite dimensional compact metric spaces and has the homotopy type of a CW-complex. Then G carries a metrizable group topology, coarser than the original one but of the same homotopy type, which turns G into an ANR-space. All these requirements are satisfied, for instance, if G itself is a countable CWcomplex. Proof. We construct a decreasing sequence of open neighborhoods Vn of the identity 1 G G such that 1. nn-c=l i n = { 1}, 2. the inclusion map Vn+1 c* Vin is null homotopic, 3. if Hm: Vm+i x I -* Vm is the null homotopy from 2, then for any g G Vm+i and each n > m there exist N > n and E > 0 such that g' E gVN, It'-tj Hm (g/, t') (E Hm (g, t) Vn, 4.Vn-1 =?Vn andVn+,Vn+l CVn, 5. for each n and every g E G there exists N > n with g-1VNg C Vn. Received by the editors April 1, 1996. 1991 Mathematics Subject Classification. Primary 54H11, 54C55, 22A05.

Detailed proofs are given in [4]. LITERATURE CITED Io V. Volterra, Theory of Functionals and of Integral and Integrodifferential Equations. Dover, New York (1959). 2. B.A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry [in Russian], Nauka, Moscow (1979). 3. L.D. Landau and E. M. Lifshits, Statistical Physics [in Russian], Part I, Nauka, Moscow (1976). 4. V.V. Vedenyapin, "Differential forms in infinite-dimensional spaces, Stokes' formula and its application to kinetic equations," Preprint, M. V. Keldysh Institute for Applied Mathematics, Academy of Sciences of the USSR, No. 137, Moscow (1983). 5. V.V. Vedenyapin, Dokl. Akad. Nauk SSSR, 233, No. 5, 765-768 (1977). 6. V.V. Vedenyapin, "On the uniqueness of Boltzmann's H-function," Preprint, Institute for Applied Mathematics, Academy of Sciences of the USSR, No. 3, Moscow (1977). 7. L. Val'dman, "The transfer phenomenon in gases under a mean pressure," in: The Thermo- dynamics of Gases [in Russian], Mashinostroenie, Moscow (1970). ACTION OF T-GROUPS ON VON NEUMANNALGEBRAS AND FACTORS WITH COUNTABLE FUNDAMENTAL GROUPS S. L. Gefter, V. Ya. Golodets, and N. I. Nessonov UDC 519.4 In this note we discuss the connection between the property T and the countability of a fundamental group (Sec. I) and the topology of the group of outer automorphisms of a type Ill factor (Sec. 2). It is shown that a T-factor [I] can arise in extending a hyperfinite lll-factor R by using a T-group (Sec. 3). I. THEOREM 1.1. Let M be a factor of type Iii with a separable predual, and N be a subfactor of M with'the property T [I]. If N'n M=C, then the fundamental group of M is countable. Theorem 1.1 is proved in exactly the same way as Theorem 2.1 in [3]. We describe only the end of the proof. To the factor Mp=K~®B (for the notation see [3, Sec. 2]) in our theorem corresponds the ll~-factor P= M®B , and to the group A G corresponds the group A N algebraically generated by IntP and {8~AutP: 8(z®i)=x®i,

Euler characteristics of 3-manifold groups and discrete subgroups of SL(2, [formula omitted

Fundamental groups of asymptotic cones

We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.

Let G be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups satisfying the strong Atiyah conjecture over a field closed under complex conjugation. Assume that the orders of finite subgroups of G are bounded above. We show that G satisfies the strong Atiyah conjecture over K. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the ∗-regular closure of K[G] in , , is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding ∗-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over K are also closed under the graph of groups construction as long as the edge groups are finite. We also infer some consequences on the structure of the K0 and K1-groups of . The techniques developed enable us to prove that K[G] fulfills the strong, algebraic and center-valued Atiyah conjectures, and that is the universal localization of K[G] over the set of all matrices that become invertible in , provided that G belongs to a certain class of groups , which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.

Notion of acyclic models are introduced in Eleinberg-Maclane [4]. In [5] and [3], this theory is used as auxiliary tools to solve extension problems of morphisms of chains complexes and homotopy between those morphisms.So in the first section of this work, we will adapt the notion of acyclic models in the category of Banach chain differential complexes Ch?(Ban). In the second section, we recall the functor of real ??-singular homology (cf. [8]) on which we apply theorems proved in the first section. In particular, we prove an analogous of Zilber-Eilenberg theorem [5] in real ??-singular homology. In last section, we prove an analogous of Brown theorem in real ??-singular homology. As consequence of this theorem we show that the real ??-singular homology depends only on the fundamental group and we establish some exact sequences.

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

We prove that every free group of nite rank can be realized as the fundamental group of a planar Rauzy fractal associated with a 4-letter unimodular cubic Pisot substitution. is characterizes all countable fundamental groups for planar Rauzy fractals. We give an explicit construction relying on two operations on substitutions: symbolic splittings and conjugations by free group automorphisms.

Contractible open 3-manifolds with free covering translation groups