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.

Computing with knot quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850074
Author(s):  
Graham Ellis ◽  
Cédric Fragnaud
Keyword(s):  
Abelian Groups ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Knot Invariant ◽  
Number Formula ◽  
Finitely Presented

The number [Formula: see text] of colorings of a knot [Formula: see text] by a finite quandle [Formula: see text] has been used in the literature to distinguish between knot types. In this paper, we suggest a refinement [Formula: see text] to this knot invariant involving any computable functor [Formula: see text] from finitely presented groups to finitely generated abelian groups. We are mainly interested in the functor [Formula: see text] that sends each finitely presented group [Formula: see text] to its abelianization [Formula: see text]. We describe algorithms needed for computing the refined invariant and illustrate implementations that have been made available as part of the HAP package for the GAP system for computational algebra. We use these implementations to investigate the performance of the refined invariant on prime knots with [Formula: see text] crossings.

scholarly journals INDICABLE GROUPS AND ENDOMORPHIC PRESENTATIONS

2011 ◽  
Vol 54 (2) ◽  
pp. 335-344
Author(s):  
MUSTAFA GÖKHAN BENLI
Keyword(s):  
Normal Subgroup ◽  
Semidirect Product ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Free Semigroups ◽  
Finitely Presented ◽  
Indicable Group

AbstractIn this paper we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if H is a finitely generated normal subgroup of a finitely presented group G with G/H cyclic, then H has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product H ⋊ ℤ, where H has finite ascending endomorphic presentation.

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.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

scholarly journals Embeddings into groups with only a few defining relations

1974 ◽  
Vol 18 (1) ◽  
pp. 1-7 ◽  
Author(s):  
W. W. Boone ◽  
D. J. Collins
Keyword(s):  
Word Problem ◽  
Finitely Presented Groups ◽  
Fixed Group ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

Sign in / Sign up

Export Citation Format

Share Document