The Existence of Root Subgroup Translated by a Given Element into its Opposite

2019 ◽  
Vol 240 (4) ◽  
pp. 494-502
Author(s):  
I. M. Pevzner
Keyword(s):  
Root Subgroup

scholarly journals Homeomorphisms of S1 and Factorization

2019 ◽  
Author(s):  
Mark Dalthorp ◽  
Doug Pickrell
Keyword(s):  
Fourier Series ◽  
Finite Type ◽  
Parameter Family ◽  
Complex Parameter ◽  
Triangular Factorization ◽  
Linear Fractional Transformations ◽  
Root Subgroup ◽  
Decay Properties ◽  
Group Of Homeomorphisms ◽  
Verblunsky Coefficients

Abstract For each $n>0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations “conjugated by $z \to z^n$”. We show that these families are free of relations, which determines the structure of “the group of homeomorphisms of finite type”. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.

Loops in SL(2, ℂ) and root subgroup factorization

2018 ◽  
Vol 07 (03) ◽  
pp. 1850008
Author(s):  
Estelle Basor ◽  
Doug Pickrell
Keyword(s):  
Critical Exponent ◽  
Toeplitz Operators ◽  
Rational Functions ◽  
Triangular Factorization ◽  
Exceptional Set ◽  
Determinant Formula ◽  
Root Subgroup ◽  
Toeplitz Determinant ◽  
Simple Generalization ◽  
Broad Outline

In previous work, we proved that for a [Formula: see text]-valued loop having the critical degree of smoothness (one half of a derivative in the [Formula: see text] Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. For a loop [Formula: see text] satisfying these conditions, the Toeplitz determinant [Formula: see text] and shifted Toeplitz determinant [Formula: see text] factor as products in root subgroup coordinates. In this paper, we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in [Formula: see text]. The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set and associated uniqueness issues, and (2) the noncompactness of [Formula: see text] entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.

Linear groups containing a root subgroup

1996 ◽  
Vol 37 (6) ◽  
pp. 1086-1100 ◽  
Author(s):  
E. L. Bashkirov
Keyword(s):  
Linear Groups ◽  
Root Subgroup
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