On the Bieri–Neumann–Strebel–Renz invariants of residually free groups

2020 ◽  
Vol 63 (3) ◽  
pp. 807-829
Author(s):  
Dessislava H. Kochloukova ◽  
Francismar Ferreira Lima
Keyword(s):  
Free Groups ◽  
Finite Union ◽  
Higher Dimensional ◽  
Homological Invariants ◽  
Finitely Presented

AbstractWe calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

scholarly journals Inversion is Possible in Groups with no Periodic Automorphisms

2015 ◽  
Vol 59 (1) ◽  
pp. 11-16
Author(s):  
Martin R. Bridson ◽  
Hamish Short
Keyword(s):  
Free Groups ◽  
Trivial Element ◽  
Finitely Presented ◽  
Torsion Free

AbstractThere exist infinite finitely presented torsion-free groups G such that Aut(G) and Out(G) are torsion free but G has an automorphism sending some non-trivial element to its inverse.

scholarly journals Abelian varieties isogenous to a power of an elliptic curve

2018 ◽  
Vol 154 (5) ◽  
pp. 934-959 ◽  
Author(s):  
Bruce W. Jordan ◽  
Allan G. Keeton ◽  
Bjorn Poonen ◽  
Eric M. Rains ◽  
Nicholas Shepherd-Barron ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Abelian Variety ◽  
Opposite Direction ◽  
Abelian Varieties ◽  
Sufficient Conditions ◽  
Higher Dimensional ◽  
Free Left ◽  
Necessary And Sufficient ◽  
Finitely Presented ◽  
Torsion Free

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

scholarly journals STABILITY, COHOMOLOGY VANISHING, AND NONAPPROXIMABLE GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MARCUS DE CHIFFRE ◽  
LEV GLEBSKY ◽  
ALEXANDER LUBOTZKY ◽  
ANDREAS THOM
Keyword(s):  
Symmetric Groups ◽  
Unitary Groups ◽  
Frobenius Norm ◽  
Central Extensions ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Open Questions ◽  
Higher Dimensional ◽  
First Time ◽  
Finitely Presented

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$ , the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

scholarly journals A periodicity theorem for acylindrically hyperbolic groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Bogopolski
Keyword(s):  
Free Groups ◽  
Hyperbolic Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Periodicity Theorem ◽  
Finitely Presented

AbstractWe generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

scholarly journals Normal amenable subgroups of the automorphism group of the full shift

2017 ◽  
Vol 39 (5) ◽  
pp. 1290-1298 ◽  
Author(s):  
JOSHUA FRISCH ◽  
TOMER SCHLANK ◽  
OMER TAMUZ
Keyword(s):  
Automorphism Group ◽  
Free Groups ◽  
Higher Dimensional ◽  
Amenable Subgroup ◽  
Full Shift ◽  
Topological Boundary

We show that every normal amenable subgroup of the automorphism group of the full shift is contained in its center. This follows from the analysis of this group’s Furstenberg topological boundary, through the construction of a minimal and strongly proximal action. We extend this result to higher dimensional full shifts. This also provides a new proof of Ryan’s theorem and of the fact that these groups contain free groups.

scholarly journals ACTIONS, LENGTH FUNCTIONS, AND NON-ARCHIMEDEAN WORDS

2013 ◽  
Vol 23 (02) ◽  
pp. 325-455 ◽  
Author(s):  
OLGA KHARLAMPOVICH ◽  
ALEXEI MYASNIKOV ◽  
DENIS SERBIN
Keyword(s):  
Free Groups ◽  
Length Functions ◽  
Methods And Techniques ◽  
Recent Developments ◽  
Finitely Presented

In this paper we survey recent developments in the theory of groups acting on Λ-trees. We are trying to unify all significant methods and techniques, both classical and recently developed, in an attempt to present various faces of the theory and to show how these methods can be used to solve major problems about finitely presented Λ-free groups. Besides surveying results known up to date we draw many new corollaries concerning structural and algorithmic properties of such groups.

scholarly journals Confluence of algebraic rewriting systems

2021 ◽  
pp. 1-28
Author(s):  
Cyrille Chenavier ◽  
Benjamin Dupont ◽  
Philippe Malbos
Keyword(s):  
Decision Problems ◽  
Algebraic Structures ◽  
Algebraic Models ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Higher Dimensional ◽  
Homological Invariants

Abstract Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

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