The orbits of the Hurwitz action of the braid groups on the standard generators

Yoshiro Yaguchi

Displaying similar documents to “The orbits of the Hurwitz action of the braid groups on the standard generators”

Non-orbit equivalent actions of $𝔽_{n}$

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

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For any $2 \leq n \leq \infty$ , we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group $𝔽_{n}$ . These actions arise as diagonal products between a generalized Bernoulli action and the action $𝔽_{n} ↷ (𝕋^{2}, λ^{2})$ , where $𝔽_{n}$ is seen as a subgroup of ${SL}_{2} (ℤ)$ .

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,...

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder, A. Veselov (2005)

Journal of the European Mathematical Society

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We consider both standard and twisted actions of a (real) Coxeter group $G$ on the complement $ℳ_{G}$ to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in $G$ and give explicit formulae which describe both actions on the total cohomology $H^{*} (ℳ_{G}, 𝒞)$ in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group $S_{n}$ , the...

Shadowing in actions of some Abelian groups

Sergei Yu. Pilyugin, Sergei B. Tikhomirov (2003)

Fundamenta Mathematicae

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We study shadowing properties of continuous actions of the groups $ℤ^{p}$ and $ℤ^{p} \times ℝ^{p}$ . Necessary and sufficient conditions are given under which a linear action of $ℤ^{p}$ on $ℂ^{m}$ has a Lipschitz shadowing property.

On the structural theory of ${II}_{1}$ factors of negatively curved groups

Ionut Chifan, Thomas Sinclair (2013)

Annales scientifiques de l'École Normale Supérieure

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Ozawa showed in [21] that for any i.c.c. hyperbolic group, the associated group factor $L Γ$ is solid. Developing a new approach that combines some methods of Peterson [29], Ozawa and Popa [27, 28], and Ozawa [25], we strengthen this result by showing that $L Γ$ is strongly solid. Using our methods in cooperation with a cocycle superrigidity result of Ioana [12], we show that profinite actions of lattices in $Sp (n, 1)$ , $n \geq 2$ , are virtually $W^{*}$ -superrigid.

On the complexity of braids

Ivan Dynnikov, Bert Wiest (2007)

Journal of the European Mathematical Society

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We define a measure of “complexity” of a braid which is natural with respect to both an algebraic and a geometric point of view. Algebraically, we modify the standard notion of the length of a braid by introducing generators $_{i j}$ , which are Garside-like half-twists involving strings $i$ through $j$ , and by counting powered generators $Δ_{i j}^{k}$ as $log (| k | + 1)$ instead of simply $| k |$ . The geometrical complexity is some natural measure of the amount of distortion of the $n$ times punctured disk caused by a homeomorphism....

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...

Invariant symbolic calculus for semidirect products

Benjamin Cahen (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be the semidirect product $V ⋊ K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$ . Let $π$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$ . We construct an invariant symbolic calculus for $π$ , under some technical hypothesis. We give some examples including the Poincaré group. ...

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.

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.

A note on normal generation and generation of groups

Andreas Thom (2015)

Communications in Mathematics

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In this note we study sets of normal generators of finitely presented residually $p$ -finite groups. We show that if an infinite, finitely presented, residually $p$ -finite group $G$ is normally generated by $g_{1}, \dots, g_{k}$ with order $n_{1}, \dots, n_{k} \in {1, 2, \dots} \cup {\infty}$ , then $β_{1}^{(2)} (G) \leq k - 1 - \sum_{i = 1}^{k} \frac{1}{n_{i}},$ where $β_{1}^{(2)} (G)$ denotes the first $ℓ^{2}$ -Betti number of $G$ . We also show that any $k$ -generated group with $β_{1}^{(2)} (G) \geq k - 1 - ε$ must have girth greater than or equal $1 / ε$ .

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $G$ be a group and $m$ be an integer greater than or equal to $2$ . $G$ is said to be $m$ -permutable if every product of $m$ elements can be reordered at least in one way. We prove that, if $G$ has a centre of finite index $z$ , then $G$ is $(1+[z/2])$ -permutable. More bounds are given on the least $m$ such that $G$ is $m$ -permutable.

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

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We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

The 4-string braid group $B_{4}$ has property RD and exponential mesoscopic rank

Sylvain Barré, Mikaël Pichot (2011)

Bulletin de la Société Mathématique de France

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We prove that the braid group $B_{4}$ on 4 strings, its central quotient $B_{4} / 〈 z 〉$ , and the automorphism group $Aut (F_{2})$ of the free group $F_{2}$ on 2 generators, have the property RD of Haagerup–Jolissaint. We also prove that the braid group $B_{4}$ is a group of intermediate mesoscopic rank (of dimension 3). More precisely, we show that the above three groups have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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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...

Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

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A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$ -flag manifold $Z = G^{ℂ} / Q$ . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...