scholarly journals $C^*$-algebras associated with the fundamental groups of graphs of groups

2005 ◽  
Vol 97 (1) ◽  
pp. 49 ◽  
Author(s):  
Rui Okayasu
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Fundamental Groups ◽  
Graph Of Groups ◽  
C Algebra ◽  
C Algebras ◽  
Graphs Of Groups ◽  
Word Hyperbolic Group ◽  
Dimensional Boundary ◽  
The Ideal

We construct a nuclear $C^*$-algebra associated with the fundamental group of a graph of groups of finite type. It is well-known that every word-hyperbolic group with zero-dimensional boundary, in other words, every group acting trees with finite stabilizers is given by the fundamental group of such a graph of groups. We show that our $C^*$-algebra is $*$-isomorphic to the crossed product arising from the associated boundary action and is also given by a Cuntz-Pimsner algebra. We also compute the K-groups and determine the ideal structures of our $C^*$-algebras.

scholarly journals Fundamental Group of Simple C*-algebras with Unique Trace III

2012 ◽  
Vol 64 (3) ◽  
pp. 573-587 ◽  
Author(s):  
Norio Nawata
Keyword(s):  
Fundamental Group ◽  
Lower Semicontinuous ◽  
Classification Theorem ◽  
Fundamental Groups ◽  
C Algebras ◽  
Scalar Multiple ◽  
Unique Trace ◽  
Densely Defined

Abstract We introduce the fundamental group ℱ(A) of a simple σ-inital C*-algebra A with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple C*-algebras with Unique Trace I and II by Nawata andWatatani. Our definition in this paper makes sense for stably projectionless C*-algebras. We show that there exist separable stably projectionless C*-algebras such that their fundamental groups are equal to ℝ×+ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Free group C*-algebras associated with ℓp

2014 ◽  
Vol 25 (07) ◽  
pp. 1450065 ◽  
Author(s):  
Rui Okayasu
Keyword(s):  
Free Group ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Linear Functionals ◽  
C Algebra ◽  
Tracial State ◽  
C Algebras ◽  
Positive Linear Functionals ◽  
The Ideal

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓp. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

ENDS OF CUSP-UNIFORM GROUPS OF LOCALLY CONNECTED CONTINUA — I

2005 ◽  
Vol 15 (04) ◽  
pp. 765-798 ◽  
Author(s):  
DAN P. GURALNIK
Keyword(s):  
Hyperbolic Group ◽  
Infinite Group ◽  
Graph Of Groups ◽  
Locally Finite ◽  
Cut Point ◽  
Word Hyperbolic Group ◽  
Complete Knowledge ◽  
Simplicial Tree ◽  
Locally Connected Continuum ◽  
Uniform Group

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia. A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove: Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers. In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

scholarly journals Surface groups within Baumslag doubles

2011 ◽  
Vol 54 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Surface Groups ◽  
Word Hyperbolic Group ◽  
Genus 2

AbstractA 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 ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

Amalgamated Products and the Howson Property

1997 ◽  
Vol 40 (3) ◽  
pp. 330-340 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Amalgamated Products ◽  
Word Hyperbolic Groups ◽  
Free Word ◽  
Torsion Free ◽  
Maximal Cyclic Subgroup

AbstractWe show that if A is a torsion-free word hyperbolic group which belongs to class (Q), that is all finitely generated subgroups of A are quasiconvex in A, then any maximal cyclic subgroup U of A is a Burns subgroup of A. This, in particular, implies that if B is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then A *UB, ⧼A, t | Ut = V⧽ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting 3-manifold groups are known to belong to class (Q) and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

C*-ALGEBRAS OF SINGULAR INTEGRAL OPERATORS AND TOEPLITZ OPERATORS ASSOCIATED WITH n-DIMENSIONAL FLOWS

1992 ◽  
Vol 03 (04) ◽  
pp. 525-579
Author(s):  
PAUL S. MUHLY ◽  
JINGBO XIA
Keyword(s):  
Singular Integral ◽  
Toeplitz Operators ◽  
Type I ◽  
Covariant Representation ◽  
Hilbert Transforms ◽  
C Algebra ◽  
C Algebras ◽  
Dimensional Dynamical System ◽  
A Chain ◽  
The Ideal

For a given covariant representation π of an n-dimensional dynamical system (X, R n, α), we study the C*-algebras [Formula: see text] of abstract singular integral opertors and Toeplitz operators generated by π(C(X)) and Hilbert transforms corresponding to n linearly independent directions. Such an algebra has a chain of ideals [Formula: see text]. We compute all [Formula: see text] and show that for 0≤k≤n−1 each of these quotients is the direct sum of C*-algebras which can be thought of as [Formula: see text] for flows of lower dimensions. The ideal structure of [Formula: see text] is carefully studied. We also determine precisely when [Formula: see text] is a C*-algebra of type I.

scholarly journals AUTOMATIC STRUCTURES AND BOUNDARIES FOR GRAPHS OF GROUPS

1994 ◽  
Vol 04 (04) ◽  
pp. 591-616 ◽  
Author(s):  
WALTER D. NEUMANN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Fundamental Group ◽  
Equivalence Relation ◽  
Infinite Product ◽  
Graph Product ◽  
Graph Of Groups ◽  
Automatic Structures ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Edge Group ◽  
Labelled Graphs

We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.

scholarly journals The Farrell-Jones isomorphism conjecture for 3-manifold groups

2007 ◽  
Vol 1 (1) ◽  
pp. 49-82 ◽  
Author(s):  
S.K. Roushon
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Partial Solution ◽  
Fundamental Groups ◽  
Problem List ◽  
Index Subgroup ◽  
Main Aspect ◽  
Graph Of Groups ◽  
Genus 2 ◽  
Finite Index Subgroup

AbstractWe show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to prove the FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy the FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby. We also deduce that the FIC is true for a class of virtually P D3-groups.Another main aspect of this article is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for B-groups. By definition a B-group contains a finite index subgroup isomorphic to the fundamental group of a compact irreducible 3-manifold with incompressible nonempty boundary so that each boundary component is of genus ≥ 2. We also prove the FIC for a large class of B-groups and moreover, using a recent result of L.E. Jones we show that the surjective part of the FIC is true for any B-group.

scholarly journals The Weak Ideal Property and Topological Dimension Zero

2017 ◽  
Vol 69 (6) ◽  
pp. 1385-1421 ◽  
Author(s):  
Cornel Pasnicu ◽  
N. Christopher Phillips
Keyword(s):  
Hausdorff Space ◽  
Crossed Products ◽  
Topological Dimension ◽  
C Algebra ◽  
Dimension Zero ◽  
C Algebras ◽  
Ideal Property ◽  
Locally Compact Hausdorff Space ◽  
Totally Disconnected ◽  
The Ideal

AbstractFollowing up on previous work, we prove a number of results for C* -algebras with the weak ideal property or topological dimension zero, and some results for C* -algebras with related properties. Some of the more important results include the following:The weak ideal property implies topological dimension zero.For a separable C* -algebra A, topological dimension zero is equivalent to , to D ⊗ A having the ideal property for some (or any) Kirchberg algebra D, and to A being residually hereditarily in the class of all C* -algebras B such that contains a nonzero projection.Extending the known result for , the classes of C* -algebras with residual (SP), which are residually hereditarily (properly) infinite, or which are purely infinite and have the ideal property, are closed under crossed products by arbitrary actions of abelian 2-groups.If A and B are separable, one of them is exact, A has the ideal property, and B has the weak ideal property, then A ⊗ B has the weak ideal property.If X is a totally disconnected locally compact Hausdorff space and A is a C0(X)-algebra all of whose fibers have one of the weak ideal property, topological dimension zero, residual (SP), or the combination of pure infiniteness and the ideal property, then A also has the corresponding property (for topological dimension zero, provided A is separable).Topological dimension zero, the weak ideal property, and the ideal property are all equivalent for a substantial class of separable C* -algebras, including all separable locally AH algebras.The weak ideal property does not imply the ideal property for separable Z-stable C* -algebras.We give other related results, as well as counterexamples to several other statements one might conjecture.

scholarly journals C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

2020 ◽  
pp. 1-46 ◽  
Author(s):  
SERGEY BEZUGLYI ◽  
ZHUANG NIU ◽  
WEI SUN
Keyword(s):  
Directed Graphs ◽  
Cantor Set ◽  
Stable Rank ◽  
Crossed Product ◽  
Real Rank ◽  
C Algebra ◽  
C Algebras ◽  
Rank 2 ◽  
Stably Finite ◽  
The Ideal

We study homeomorphisms of a Cantor set with $k$ ( $k<+\infty$ ) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$ , the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

Sign in / Sign up

Export Citation Format

Share Document