Stackings and the W-cycles Conjecture

AbstractWe prove Wise’s W-cycles conjecture. Consider a compact graph Γ' immersing into another graph Γ. For any immersed cycle Λ: S1 ⟶ Γ, we consider the map Λ' from the circular components 𝕊 of the pullback to Γ'. Unless Λ' is reducible, the degree of the covering map 𝕊 ⟶ S1 is bounded above by minus the Euler characteristic of Γ'. As a corollary, any finitely generated subgroup of a one-relator group has a finitely generated Schur multiplier.

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EULER CHARACTERISTICS AS INVARIANTS OF IWASAWA MODULES

Proceedings of the London Mathematical Society ◽

10.1112/s0024611502013680 ◽

2002 ◽

Vol 85 (3) ◽

pp. 634-658 ◽

Cited By ~ 21

Author(s):

SUSAN HOWSON

Keyword(s):

Elliptic Curves ◽

Euler Characteristic ◽

Lie Group ◽

Iwasawa Theory ◽

Subject Classification ◽

Finitely Generated ◽

Euler Characteristics ◽

Primary 16E10 ◽

Iwasawa Algebra

If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves.2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

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The Euler characteristic of a finitely generated module of finite injective dimension

Mathematische Zeitschrift ◽

10.1007/bf01178979 ◽

1973 ◽

Vol 130 (1) ◽

pp. 79-93 ◽

Cited By ~ 8

Author(s):

Rodney Y. Sharp

Keyword(s):

Euler Characteristic ◽

Finitely Generated ◽

Injective Dimension ◽

Finitely Generated Module

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An Euler characteristic for modules of finite G-dimension

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15059 ◽

2008 ◽

Vol 102 (2) ◽

pp. 206 ◽

Cited By ~ 2

Author(s):

Sean Sather-Wagstaff ◽

Diana White

Keyword(s):

Euler Characteristic ◽

Betti Numbers ◽

Projective Dimension ◽

Finitely Generated ◽

Finite Projective Dimension ◽

Extremal Behavior ◽

Classical Invariant ◽

Category Of Modules ◽

Finitely Generated Modules

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.

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Ends of spaces related by a covering map

Canadian Mathematical Bulletin ◽

10.4153/cmb-1990-019-2 ◽

1990 ◽

Vol 33 (1) ◽

pp. 110-118

Author(s):

Georg peschke

Keyword(s):

Classical Result ◽

Topological Spaces ◽

Finitely Generated ◽

Finitely Generated Group ◽

Covering Map ◽

End Space ◽

Compact Polyhedron

Consider a covering p : X → B of connected topological spaces. If B is a compact polyhedron, a classical result of H. Hopf [4] says that the end space E(X) of X is an invariant of the group G of covering transformations. Thus it becomes meaningful to define the end space of the finitely generated group G as E(G) := E(X).

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Schur's theorem and its relation to the closure properties of the non-abelian tensor product

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.150 ◽

2019 ◽

Vol 150 (2) ◽

pp. 993-1002

Author(s):

G. Donadze ◽

M. Ladra ◽

P. Páez-Guillán

Keyword(s):

Finite Group ◽

Tensor Product ◽

Schur Multiplier ◽

Finitely Generated ◽

Closure Properties ◽

Schur’S Theorem

AbstractWe show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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