scholarly journals An alternate proof that the fundamental group of a Peano continuum is finitely presented if the group is countable

2011 ◽  
Vol 46 (2) ◽  
pp. 505-511 ◽  
Author(s):  
J. Dydak ◽  
Z Virk
Keyword(s):  
Fundamental Group ◽  
Alternate Proof ◽  
Peano Continuum ◽  
Finitely Presented

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.

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.

Representations of Subgroups of Universal Triangle Groups

2007 ◽  
Vol 14 (02) ◽  
pp. 181-190 ◽  
Author(s):  
Jianguo Xia
Keyword(s):  
Fundamental Group ◽  
Chamber System ◽  
Triangle Group ◽  
Triangle Groups ◽  
Triangle Geometry ◽  
Finitely Presented Group ◽  
Finitely Presented

Let G be a universal triangle group, and H a subgroup of G such that the chamber system ΔH is a tight triangle geometry. Then H, which is canonically isomorphic to the topological fundamental group π1(ΔH) of ΔH, is a finitely presented group. For some H we give their representations.

scholarly journals Group extensions are quasi-simply-filtrated

1994 ◽  
Vol 50 (1) ◽  
pp. 21-27 ◽  
Author(s):  
S.G. Brick ◽  
M.L. Mihalik
Keyword(s):  
Fundamental Group ◽  
Universal Cover ◽  
Infinite Group ◽  
Finite Complex ◽  
Group Extensions ◽  
Finitely Presented Group ◽  
Simply Connected ◽  
Finitely Presented

A finitely presented group G is quasi-simply-filtrated (abbreviated qsf) if, given a finite complex with fundamental group G, the universal cover of the complex can be “approximated” by simply connected finite complexes. This notion is a generalisation of a concept of Casson's used in the study of three-manifolds.In this paper we show that any extension of a finitely presented infinite group by a finitely presented infinite group is qsf.

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.

Cockcroft Properties of Thompson’s Group

2000 ◽  
Vol 43 (3) ◽  
pp. 268-281
Author(s):  
W. A. Bogley ◽  
N. D. Gilbert ◽  
James Howie
Keyword(s):  
Fundamental Group ◽  
Euler Characteristic ◽  
Word Problem ◽  
General Result ◽  
Cantor Set ◽  
Regular Covering ◽  
Thompson’S Group ◽  
Finitely Presented ◽  
Proper Covering

AbstractIn a study of the word problem for groups, R. J. Thompson considered a certain group F of self-homeomorphisms of the Cantor set and showed, among other things, that F is finitely presented. Using results of K. S. Brown and R. Geoghegan, M. N. Dyer showed that F is the fundamental group of a finite two-complex Z2 having Euler characteristic one and which is Cockcroft, in the sense that each map of the two-sphere into Z2 is homologically trivial. We show that no proper covering complex of Z2 is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group F is Cockcroft.

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.

scholarly journals Planar Transitive Graphs

10.37236/7888 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Matthias Hamann
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Transitive Graph ◽  
Locally Finite ◽  
Transitive Graphs ◽  
Finitely Presented

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible. 

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.

scholarly journals Hopfian and co-Hopfian groups

1997 ◽  
Vol 56 (1) ◽  
pp. 17-24 ◽  
Author(s):  
Satya Deo ◽  
K. Varadarajan
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Aspherical Manifold ◽  
Finitely Presented

The main results proved in this note are the following:(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and co-Hopfian.(ii) There is no functorial imbedding of groups (respectively finitely generated groups) into Hopfian groups.(iii) We prove a result which implies in particular that if the double orientable cover N of a closed non-orientable aspherical manifold M has a co-Hopfian fundamental group then π1(M) itself is co-Hopfian.

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