Statistical Properties of Locally Free Groups�with Applications to Braid Groups�and Growth of Random Heaps

2000 ◽  
Vol 212 (2) ◽  
pp. 469-501 ◽  
Author(s):  
A. M. Vershik ◽  
S. Nechaev ◽  
R. Bikbov
Keyword(s):  
Free Groups ◽  
Braid Groups ◽  
Statistical Properties

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Free groups and finite-type invariants of pure braids

2002 ◽  
Vol 132 (1) ◽  
pp. 117-130 ◽  
Author(s):  
JACOB MOSTOVOY ◽  
SIMON WILLERTON
Keyword(s):  
Finite Type ◽  
Free Group ◽  
Free Groups ◽  
Braid Groups ◽  
Point Of View ◽  
Magnus Expansion ◽  
Vassiliev Invariants ◽  
Integer Coefficients ◽  
Finite Type Invariants ◽  
Theoretic Point

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

An infinite family of braid group representations in automorphism groups of free groups

2020 ◽  
Vol 29 (10) ◽  
pp. 2042007
Author(s):  
Wonjun Chang ◽  
Byung Chun Kim ◽  
Yongjin Song
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Infinite Family ◽  
Braid Groups ◽  
Mapping Class ◽  
Group Representations ◽  
Class Groups ◽  
Branched Coverings

The [Formula: see text]-fold ([Formula: see text]) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator [Formula: see text] of braid group inside mapping class group induced by [Formula: see text]-fold covering is the product of [Formula: see text] Dehn twists on the surface.

Pure braid groups and products of free groups

1988 ◽  
pp. 217-228 ◽  
Author(s):  
Michael Falk ◽  
Richard Randell
Keyword(s):  
Free Groups ◽  
Braid Groups

On representations of Artin–Tits and surface braid groups

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Valeriy G. Bardakov ◽  
Paolo Bellingeri
Keyword(s):  
Free Groups ◽  
Braid Groups ◽  
Finitely Generated ◽  
Induced Representations ◽  
Automorphisms Of Free Groups ◽  
Closed Surfaces ◽  
Outer Automorphisms

AbstractWe define and study extensions of the Artin and Perron–Vannier representations of braid groups to topological and algebraic generalizations of braid groups. We provide faithful representations of braid groups of oriented surfaces with boundary as automorphisms of finitely generated free groups. The induced representations of such groups as outer automorphisms of finitely generated free groups are still faithful. Also we give a representation of braid groups of closed surfaces as outer automorphisms of finitely generated free groups. Finally, we provide faithful representations of Artin–Tits groups of type 𝒟 as automorphisms of free groups.

scholarly journals Statistical properties of subgroups of free groups

2012 ◽  
Vol 42 (3) ◽  
pp. 349-373
Author(s):  
Frédérique Bassino ◽  
Armando Martino ◽  
Cyril Nicaud ◽  
Enric Ventura ◽  
Pascal Weil
Keyword(s):  
Free Groups ◽  
Statistical Properties

Office Hours with a Geometric Group Theorist

2017 ◽  
Keyword(s):  
Large Scale ◽  
Coxeter Groups ◽  
Free Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Gromov Hyperbolicity ◽  
Geometric Invariants ◽  
Automorphisms Of Free Groups ◽  
Thompson's Groups ◽  
Ping Pong

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. This book brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. It features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

scholarly journals An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

2003 ◽  
Vol 55 (4) ◽  
pp. 822-838 ◽  
Author(s):  
Djun Maximilian Kim ◽  
Dale Rolfsen
Keyword(s):  
Fibre Type ◽  
Free Groups ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
The Other ◽  
Fundamental Groups ◽  
Magnus Expansion ◽  
Direct Generalization ◽  
Complex Hyperplane

AbstractWe define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

On braids and groups Gnk

2015 ◽  
Vol 24 (13) ◽  
pp. 1541009 ◽  
Author(s):  
Vassily Olegovich Manturov ◽  
Igor Mikhailovich Nikonov
Keyword(s):  
Dynamical Systems ◽  
Braid Group ◽  
Knot Theory ◽  
Free Groups ◽  
Braid Groups ◽  
The Other ◽  
Pure Braid Group ◽  
And Topology ◽  
Other Hand ◽  
Definition Of

In [Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208.] the first author gave the definition of [Formula: see text]-free braid groups [Formula: see text]. Here we establish connections between free braid groups, classical braid groups and free groups: we describe explicitly the homomorphism from (pure) braid group to [Formula: see text]-free braid groups for important cases [Formula: see text]. On the other hand, we construct a homomorphism from (a subgroup of) free braid groups to free groups. The relations established would allow one to construct new invariants of braids and to define new powerful and easily calculated complexities for classical braid groups.

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