scholarly journals Coarse fundamental groups and box spaces

2019 ◽  
Vol 150 (3) ◽  
pp. 1139-1154
Author(s):  
Thiebout Delabie ◽  
Ana Khukhro
Keyword(s):  
Fundamental Group ◽  
Betti Number ◽  
Equivalence Classes ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Finitely Presented Groups ◽  
Rank Gradient ◽  
Box Space ◽  
Finitely Presented ◽  
Uniformly Bounded

AbstractWe use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

Bounded Depth Ascending HNN Extensions and -Semistability at infinity

2019 ◽  
Vol 72 (6) ◽  
pp. 1529-1550
Author(s):  
Michael L. Mihalik
Keyword(s):  
Fundamental Group ◽  
Companion Paper ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
New Methods ◽  
Hnn Extensions ◽  
Finitely Presented

AbstractA well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class ${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class ${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.

Non-cocompact Group Actions and -Semistability at Infinity

2019 ◽  
Vol 72 (5) ◽  
pp. 1275-1303 ◽  
Author(s):  
Ross Geoghegan ◽  
Craig Guilbault ◽  
Michael Mihalik
Keyword(s):  
Fundamental Group ◽  
Group Actions ◽  
Fundamental Property ◽  
Companion Paper ◽  
Infinite Index ◽  
Locally Compact ◽  
Finitely Presented Groups ◽  
Simply Connected ◽  
Open Question ◽  
Finitely Presented

AbstractA finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

Elementary amenable groups and 4-manifolds with Euler characteristic 0

1991 ◽  
Vol 50 (1) ◽  
pp. 160-170 ◽  
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Normal Subgroup ◽  
Fundamental Group ◽  
Euler Characteristic ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Euler Characteristics ◽  
Amenable Groups ◽  
Amenable Subgroup ◽  
Finite Subgroups ◽  
Torsion Free

AbstractWe extend earlier work relating asphericity and Euler characteristics for finite complexes whose fundamental groups have nontrivial torsion free abelian normal subgroups. In particular a finitely presentable group which has a nontrivial elementary amenable subgroup whose finite subgroups have bounded order and with no nontrivial finite normal subgroup must have deficiency at most 1, and if it has a presentation of deficiency 1 then the corresponding 2-complex is aspherical. Similarly if the fundamental group of a closed 4-manifold with Euler characteristic 0 is virtually torsion free and elementary amenable then it either has 2 ends or is virtually an extension of Z by a subgroup of Q, or the manifold is asphencal and the group is virtually poly- Z of Hirsch length 4.

scholarly journals Nilpotent quotients of fundamental groups of special 3-manifolds with boundary

1998 ◽  
Vol 58 (2) ◽  
pp. 233-237
Author(s):  
Gabriela Putinar
Keyword(s):  
Fundamental Group ◽  
Betti Number ◽  
Fundamental Groups ◽  
Manifolds With Boundary ◽  
Manifold With Boundary ◽  
Maximal Torsion ◽  
Torsion Free

We use a Betti number estimate of Freedman-Hain-Teichner to show that the maximal torsion-free nilpotent quotient of the fundamental group of a 3-manifold with boundary is either Z or Z ⊕ Z. In particular we reobtain the Evans-Moser classification of 3-manifolds with boundary which have nilpotent fundamental groups.

FUNDAMENTAL GROUPS AND PRESENTATIONS OF ALGEBRAS

2006 ◽  
Vol 05 (05) ◽  
pp. 549-562 ◽  
Author(s):  
JUAN CARLOS BUSTAMANTE ◽  
DIANE CASTONGUAY
Keyword(s):  
Fundamental Groups ◽  
Finitely Presented Groups ◽  
Finitely Presented

In this note, we investigate how different fundamental groups of presentations of a fixed algebra A can be. For finitely many finitely presented groups Gi, we construct an algebra A such that all Gi appear as fundamental groups of presentations of A.

Kähler groups and rigidity phenomena

1991 ◽  
Vol 109 (1) ◽  
pp. 31-44 ◽  
Author(s):  
F. E. A. Johnson ◽  
E. G. Rees
Keyword(s):  
Kähler Manifolds ◽  
Fundamental Groups ◽  
Kahler Manifolds ◽  
Projective Varieties ◽  
Group Extensions ◽  
Finitely Presented Groups ◽  
Kähler Groups ◽  
Complex Projective ◽  
Finitely Presented

The class of fundamental groups of non-singular complex projective varieties is an interesting, but as yet imperfectly understood, class of finitely presented groups. Membership of is known to be extremely restricted (see [22, 23]). In this paper, we employ geometrical rigidity properties to realize some group extensions as elements of as in our previous papers, we find it convenient to work simultaneously with the class ℋ of fundamental groups of compact Kähler manifolds.

On groups and 3-manifolds with weak dimension ≤ 1

1974 ◽  
Vol 75 (1) ◽  
pp. 33-35
Author(s):  
David E. Galewski
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finite Generation ◽  
Connected Sum ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Weak Dimension ◽  
Finitely Presented

0. Introduction. A group π has weak dimension (wd) ≤ n (see Cartan and Ellen-berg (2)) if Hk(π, A) = 0 for all right Z(π)-modules A and all k > n. We say that the weak dimension of a manifold M is ≤ n if wd (πl(M))≤ n. In section 1 we show that open, orientable, irreducible 3-manifolds have wd ≤ 1 if and only if they are the monotone on of 1-handle bodies. In his celebrated theorem (10), Stallings proves that finitely presented groups of cohomological dimensions ≤ 1 are free. In section 2 we prove that if π is a finitely presented group which is the fundamental group of any orientable 3-manifold with wd ≤ 1 then π is free. We also give an example to show that the finite generation of π is necessary. (Swan (11) removes the finitely presented hypothesis from Stalling's theorem.) Finally, in section 3 we generalize a theorem of McMillan (5) and prove that if M is an open, orientable, irreducible 3-manifold with finitely generated fundamental group, then M is stably (taking the product with n ≥ 1 copies of ℝ) a connected sum along the boundary of trivial (n+2)-disc Sl bundles.

scholarly journals Reidemeister moves and groups

2015 ◽  
Vol 24 (10) ◽  
pp. 1540006
Author(s):  
Vassily Olegovich Manturov
Keyword(s):  
Group Theory ◽  
Knot Theory ◽  
Free Groups ◽  
Equivalence Classes ◽  
Finitely Presented Groups ◽  
Combinatorial Objects ◽  
Reidemeister Moves ◽  
Interesting Class ◽  
Finitely Presented

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

scholarly journals Homotopical height

2014 ◽  
Vol 25 (13) ◽  
pp. 1450123 ◽  
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Dishant Pancholi
Keyword(s):  
Fundamental Group ◽  
Group Cohomology ◽  
Point Of View ◽  
Complex Manifolds ◽  
Almost Complex Manifolds ◽  
Finitely Presented Groups ◽  
Almost Complex ◽  
Kähler Groups ◽  
Closed Contact ◽  
Finitely Presented

Given a group G and a class of manifolds 𝒞 (e.g. symplectic, contact, Kähler, etc.), it is an old problem to find a manifold MG ∈ 𝒞 whose fundamental group is G. This article refines it: for a group G and a positive integer r find MG ∈ 𝒞 such that π1(MG) = G and πi(MG) = 0 for 1 < i < r. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest r for which such an MG ∈ 𝒞 can be found is called the homotopical height ht 𝒞(G). Homotopical height provides a dimensional obstruction to finding a K(G, 1) space within the given class 𝒞, leading to a hierarchy of these classes in terms of "softness" or "hardness" à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes 𝒮𝒫 and 𝒞𝒜 of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group G can be realized as the fundamental group of a manifold in 𝒮𝒫 and a manifold in 𝒞𝒜. For these classes, ht 𝒞(G) provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP.

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