scholarly journals Arithmetic of character varieties of free groups

2015 ◽  
Vol 26 (12) ◽  
pp. 1550100 ◽  
Author(s):  
Sergey Mozgovoy ◽  
Markus Reineke
Keyword(s):  
Finite Fields ◽  
Finite Index ◽  
Free Groups ◽  
Euler Characteristics ◽  
Explicit Formulas ◽  
Character Varieties ◽  
Positivity Property

Via counting over finite fields, we derive explicit formulas for the [Formula: see text]-polynomials and Euler characteristics of [Formula: see text]- and [Formula: see text]-character varieties of free groups. We prove a positivity property for these polynomials and relate them to the number of subgroups of finite index.

Subgroups of Finite Index in Free Groups

1949 ◽  
Vol 1 (2) ◽  
pp. 187-190 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Recursion Formula ◽  
Independent Interest ◽  
Finiteness Conditions ◽  
Total Length ◽  
Free Generators

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

scholarly journals Centralisers of Dehn twist automorphisms of free groups

2015 ◽  
Vol 159 (1) ◽  
pp. 89-114 ◽  
Author(s):  
MORITZ RODENHAUSEN ◽  
RICHARD D. WADE
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Dehn Twist ◽  
Classifying Space ◽  
Dehn Twists ◽  
Automorphisms Of Free Groups

AbstractWe refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

QUADRATIC FORMULAS FOR GENERALIZED QUATERNIONS

2009 ◽  
Vol 08 (03) ◽  
pp. 289-306 ◽  
Author(s):  
MARCO ABRATE
Keyword(s):  
Finite Fields ◽  
Quaternion Algebra ◽  
Quadratic Polynomial ◽  
Explicit Formulas ◽  
Quaternion Algebras ◽  
Solving Equations ◽  
Generalized Quaternion

In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.

scholarly journals Serre polynomials of SLn- and PGLn-character varieties of free groups

2021 ◽  
Vol 161 ◽  
pp. 104008
Author(s):  
Carlos Florentino ◽  
Azizeh Nozad ◽  
Alfonso Zamora
Keyword(s):  
Free Groups ◽  
Character Varieties

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

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