Amenability and unique ergodicity of automorphism groups of Fraïssé structures

Andy Zucker

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Displaying similar documents to “Amenability and unique ergodicity of automorphism groups of Fraïssé structures”

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

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We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M (G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M (G_{)}$ . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M (G_{)}$ admits...

Polycyclic groups with automorphisms of order four

Tao Xu, Fang Zhou, Heguo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $α$ is an automorphism of order four of a polycyclic group $G$ and the map $ϕ : G \to G$ defined by $g^{ϕ} = [g, α]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{''}$ is included in the centre of $H$ and $C_{H} (α^{2})$ is abelian, both $C_{G} (α^{2})$ and $G / [G, α^{2}]$ are abelian-by-finite. These results extend recent and classical results in...

Products of conjugacy classes in finite unitary groups $G U (3, q^{2})$ and $S U (3, q^{2})$

S.Yu. Orevkov (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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For the groups $G U (3)$ , $S U (3)$ , $G L (3)$ , $S U (3)$ over a finite field we solve the class product problem, i.e., we give a complete list of $m$ -tuples of conjugacy classes whose product does not contain the identity matrix.

On automorphisms of the Banach space $ℓ_{\infty} / c ₀$

Piotr Koszmider, Cristóbal Rodríguez-Porras (2016)

Fundamenta Mathematicae

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We investigate Banach space automorphisms $T : ℓ_{\infty} / c ₀ \to ℓ_{\infty} / c ₀$ focusing on the possibility of representing their fragments of the form $T_{B, A} : ℓ_{\infty} (A) / c ₀ (A) \to ℓ_{\infty} (B) / c ₀ (B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_{\infty} (A)$ into $ℓ_{\infty} (B)$ , infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_{\infty} / c ₀$ . We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

Explicit computations of all finite index bimodules for a family of II $_{1}$ factors

Stefaan Vaes (2008)

Annales scientifiques de l'École Normale Supérieure

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We study II $_{1}$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$ - $N$ -bimodule (in particular, every isomorphism between $M$ and $N$ ) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$ - $M$ -bimodules is identified with an extended Hecke fusion algebra,...

Automorphisms of metacyclic groups

Haimiao Chen, Yueshan Xiong, Zhongjian Zhu (2018)

Czechoslovak Mathematical Journal

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A metacyclic group $H$ can be presented as $〈 α, β : α^{n} = 1$ , $β^{m} = α^{t}$ , $β α β^{- 1} = α^{r} 〉$ for some $n$ , $m$ , $t$ , $r$ . Each endomorphism $σ$ of $H$ is determined by $σ (α) = α^{x_{1}} β^{y_{1}}$ , $σ (β) = α^{x_{2}} β^{y_{2}}$ for some integers $x_{1}$ , $x_{2}$ , $y_{1}$ , $y_{2}$ . We give sufficient and necessary conditions on $x_{1}$ , $x_{2}$ , $y_{1}$ , $y_{2}$ for $σ$ to be an automorphism.

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

Finite groups whose all proper subgroups are $𝒞$ -groups

Pengfei Guo, Jianjun Liu (2018)

Czechoslovak Mathematical Journal

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A group $G$ is said to be a $𝒞$ -group if for every divisor $d$ of the order of $G$ , there exists a subgroup $H$ of $G$ of order $d$ such that $H$ is normal or abnormal in $G$ . We give a complete classification of those groups which are not $𝒞$ -groups but all of whose proper subgroups are $𝒞$ -groups.

Horocyclic products of trees

Laurent Bartholdi, Markus Neuhauser, Wolfgang Woess (2008)

Journal of the European Mathematical Society

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Let $T_{1}, \dots, T_{d}$ be homogeneous trees with degrees $q_{1} + 1, \dots, q_{d} + 1 \geq 3$ , respectively. For each tree, let $𝔥 : T_{j} \to ℤ$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_{1}, \dots, T_{d}$ is the graph $𝖣𝖫 (q_{1}, \dots, q_{d})$ consisting of all $d$ -tuples $x_{1} \dots x_{d} \in T_{1} \times \dots \times T_{d}$ with $𝔥 (x_{1}) + \dots + 𝔥 (x_{d}) = 0$ , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d = 2$ and $q_{1} = q_{2} = q$ then we obtain a Cayley graph...

The number of conjugacy classes of elements of the Cremona group of some given finite order

Jérémy Blanc (2007)

Bulletin de la Société Mathématique de France

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This note presents the study of the conjugacy classes of elements of some given finite order $n$ in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if $n$ is even, $n = 3$ or $n = 5$ , and that it is equal to $3$ (respectively $9$ ) if $n = 9$ (respectively if $n = 15$ ) and to $1$ for all remaining odd orders. Some precise representative elements of the classes are given.

Bicyclic commutator quotients with one non-elementary component

Daniel Mayer (2023)

Mathematica Bohemica

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For any number field $K$ with non-elementary $3$ -class group ${Cl}_{3} (K) ≃ C_{3^{e}} \times C_{3}$ , $e \geq 2$ , the punctured capitulation type $ϰ (K)$ of $K$ in its unramified cyclic cubic extensions $L_{i}$ , $1 \leq i \leq 4$ , is an orbit under the action of $S_{3} \times S_{3}$ . By means of Artin’s reciprocity law, the arithmetical invariant $ϰ (K)$ is translated to the punctured transfer kernel type $ϰ (G_{2})$ of the automorphism group $G_{2} = Gal (F_{3}^{2} (K) / K)$ of the second Hilbert $3$ -class field of $K$ . A classification of finite $3$ -groups $G$ with low order and bicyclic commutator quotient $G / G^{'} ≃ C_{3^{e}} \times C_{3}$ , $2 \leq e \leq 6$ , according to the algebraic...

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p = tp (b / A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd (A)$ , (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ ...

The distribution of second $p$ -class groups on coclass graphs

Daniel C. Mayer (2013)

Journal de Théorie des Nombres de Bordeaux

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General concepts and strategies are developed for identifying the isomorphism type of the second $p$ -class group $G = Gal (F_{p}^{2} (K) | K)$ , that is the Galois group of the second Hilbert $p$ -class field $F_{p}^{2} (K)$ , of a number field $K$ , for a prime $p$ . The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢 (p, r)$ , $r \geq 0$ , in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$ -class group ${Cl}_{p} (K)$ , the position of $G$ is restricted to certain admissible branches...

Fraïssé structures and a conjecture of Furstenberg

Dana Bartošová, Andy Zucker (2019)

Commentationes Mathematicae Universitatis Carolinae

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We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S (G)$ , the Samuel compactification, and $E (M (G))$ , the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of $G = S_{\infty}$ , leading us to define and investigate several...

Principalization algorithm via class group structure

Daniel C. Mayer (2014)

Journal de Théorie des Nombres de Bordeaux

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For an algebraic number field $K$ with $3$ -class group ${Cl}_{3} (K)$ of type $(3, 3)$ , the structure of the $3$ -class groups ${Cl}_{3} (N_{i})$ of the four unramified cyclic cubic extension fields $N_{i}$ , $1 \leq i \leq 4$ , of $K$ is calculated with the aid of presentations for the metabelian Galois group $G_{3}^{2} (K) = Gal (F_{3}^{2} (K) | K)$ of the second Hilbert $3$ -class field $F_{3}^{2} (K)$ of $K$ . In the case of a quadratic base field $K = ℚ (\sqrt{D})$ it is shown that the structure of the $3$ -class groups of the four $S_{3}$ -fields $N_{1}, ..., N_{4}$ frequently determines the type of principalization of the $3$ -class group of $K$ in $N_{1}, ..., N_{4}$ . This...

A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${(α_{j})}_{j = 1}^{\infty}$ of scalars, there exists a subsequence ${(α_{k_{j}})}_{j = 1}^{\infty}$ such that either every subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series, or no subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series. In particular examples we decide which of the two cases holds.