scholarly journals Universal minimal flows of automorphism groups

2003 ◽  
Vol 127 (28) ◽  
pp. 93-106
Author(s):  
A.S. Kechris ◽  
V. Pestov ◽  
S. Todorcevic
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Automorphism Groups ◽  
Ramsey Theory ◽  
Topological Dynamics ◽  
Dimensional Vector ◽  
Topological Groups ◽  
Free Graph ◽  
Dimensional Vector Space ◽  
Atomless Boolean Algebra

We investigate some connections between the Fraiss? theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As result we compute universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the ?-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory, the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.

scholarly journals On automorphism groups of low complexity subshifts

2015 ◽  
Vol 36 (1) ◽  
pp. 64-95 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS ◽  
SAMUEL PETITE
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Polynomial Complexity ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Topological Dynamics ◽  
Main Technique ◽  
Shift Map ◽  
The One ◽  
A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

scholarly journals Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups

2005 ◽  
Vol 15 (1) ◽  
pp. 106-189 ◽  
Author(s):  
A. S. Kechris ◽  
V. G. Pestov ◽  
S. Todorcevic
Keyword(s):  
Automorphism Groups ◽  
Ramsey Theory ◽  
Topological Dynamics

Combinatorial and recursive aspects of the automorphism group of the countable atomless Boolean algebra

1986 ◽  
Vol 51 (2) ◽  
pp. 292-301
Author(s):  
E. W. Madison ◽  
B. Zimmermann-Huisgen
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Recursive Function ◽  
Conjugacy Classes ◽  
Explicit Description ◽  
Free Basis ◽  
Atomless Boolean Algebra ◽  
Cycle Structures ◽  
Countably Infinite

AbstractGiven an admissible indexing φ of the countable atomless Boolean algebra ℬ, an automorphism F of ℬ is said to be recursively presented (relative to φ) if there exists a recursive function p ϵ Sym(ω) such that F ∘ φ = φ ∘ p. Our key result on recursiveness: Both the subset of Aut(ℬ) consisting of all those automorphisms which are recursively presented relative to some indexing, and its complement, the set of all “totally nonrecursive” automorphisms, are uncountable.This arises as a consequence of the following combinatorial investigations: (1) A comparison of the cycle structures of ƒ and , where ƒ is a permutation of some free basis of ℬ and is the automorphism of ℬ induced by ƒ.(2) An explicit description of the permutations of ω whose conjugacy classes in Sym(ω) are (a) uncountable, (b) countably infinite, and (c) finite.

Structure Polynomials and Subgraphs of Rooted Regular Trees

2018 ◽  
Vol 25 (01) ◽  
pp. 45-70
Author(s):  
Daniele D’Angeli ◽  
Alfredo Donno
Keyword(s):  
Automorphism Group ◽  
Rooted Tree ◽  
Equivalence Classes ◽  
Vector Subspace ◽  
Dimensional Vector ◽  
Gelfand Pairs ◽  
Dimensional Vector Space ◽  
Graph Interpretation ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field 𝔽p, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.

scholarly journals Poset Homology of Rees Products, and $q$-Eulerian Polynomials

10.37236/86 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
John Shareshian ◽  
Michelle L. Wachs
Keyword(s):  
Boolean Algebra ◽  
Vector Space ◽  
Commutative Algebra ◽  
Coxeter Groups ◽  
Dimensional Vector ◽  
Eulerian Numbers ◽  
Dimensional Vector Space ◽  
Eulerian Polynomials ◽  
Poset Homology ◽  
Poset Topology

The notion of Rees product of posets was introduced by Björner and Welker in [8], where they study connections between poset topology and commutative algebra. Björner and Welker conjectured and Jonsson [25] proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak{ S}$$_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the Boolean algebra by the lattice of subspaces of the $n$-dimensional vector space over the $q$ element field, and involves the (maj,exc)-$q$-Eulerian polynomials studied in previous papers of the authors [32,33]. Equivariant versions of the refinement and the original conjecture are also proved, as are type BC versions (in the sense of Coxeter groups) of the original conjecture and its $q$-analog.

scholarly journals Morita’s trace maps on the group of homology cobordisms

2018 ◽  
Vol 12 (03) ◽  
pp. 775-818 ◽  
Author(s):  
Gwénaël Massuyeau ◽  
Takuya Sakasai
Keyword(s):  
Automorphism Group ◽  
Class Group ◽  
Explicit Construction ◽  
Mapping Class ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Fundamental Properties ◽  
Magnus Representation ◽  
Trace Maps

Morita introduced in 2008 a [Formula: see text]-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His [Formula: see text]-cocycle contains all the “traces” of Johnson homomorphisms which he introduced 15 years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita’s [Formula: see text]-cocycle based on a simple and explicit construction. Our [Formula: see text]-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group.

Automorphism Groups of Denumerable Boolean Algebras

1977 ◽  
Vol 29 (3) ◽  
pp. 466-471 ◽  
Author(s):  
Ralph Mckenzie
Keyword(s):  
Automorphism Group ◽  
Boolean Algebra ◽  
Automorphism Groups ◽  
Boolean Algebras ◽  
Group Of Automorphisms

We are concerned with the extent to which the structure of a Boolean algebra (or BA, for brevity) is reflected in its group of automorphisms, Aut . In particular, for which algebras can one conclude that if Aut Aut , then Monk has conjectured [3] that this implication holds for denumerable BA's with at least one atom. We shall refute his conjecture, but show that the implication does hold if and are denumerable, if each has at least one atom, and if the sum of the atoms exists in . In fact, under those assumptions the algebra 21 can be rather neatly recovered from its abstract automorphism group.

scholarly journals Binary Hamming codes and Boolean designs

2021 ◽  
Author(s):  
Giovanni Falcone ◽  
Marco Pavone
Keyword(s):  
Weight Distribution ◽  
Automorphism Groups ◽  
Galois Field ◽  
Hamming Weight ◽  
Dimensional Vector ◽  
Hamming Code ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
The Family

AbstractIn this paper we consider a finite-dimensional vector space $${\mathcal {P}}$$ P over the Galois field $${\text {GF}}(2),$$ GF ( 2 ) , and the family $${\mathcal {B}}_k$$ B k (respectively, $${\mathcal {B}}_k^*$$ B k ∗ ) of all the k-sets of elements of $$\mathcal {P}$$ P (respectively, of $${\mathcal {P}}^*= {\mathcal {P}} \setminus \{0\}$$ P ∗ = P \ { 0 } ) summing up to zero. We compute the parameters of the 3-design $$({\mathcal {P}},{\mathcal {B}}_k)$$ ( P , B k ) for any (necessarily even) k, and of the 2-design $$({\mathcal {P}}^{*},{\mathcal {B}}_k^{*})$$ ( P ∗ , B k ∗ ) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of $${\mathcal {P}}$$ P , respectively of $${\mathcal {P}}^*$$ P ∗ , that induce permutations of $${\mathcal {B}}_k$$ B k , respectively of $${\mathcal {B}}_k^*.$$ B k ∗ . In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

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