Free Group Automorphisms and Knotted Tori in S4

1997 ◽  
Vol 06 (01) ◽  
pp. 95-103 ◽  
Author(s):  
Daniel S. Silver
Keyword(s):  
Free Group ◽  
Commutator Subgroup ◽  
Sufficient Conditions ◽  
Simple Construction ◽  
Finitely Generated ◽  
Seifert Manifolds ◽  
Group Automorphisms ◽  
Free Group Automorphisms ◽  
Finitely Presented

Let ϕ be an automorphism of a finitely generated free group F, w and element of F, and H the subgroup of F generated by the orbit {ϕn (w), |n ∈ Z}. We describe sufficient conditions ensuring that H is non-finitely generated. Using this we give a simple construction of tori T embedded in S4 in such a way that the commutator subgroup of π1(S4 - T) is finitely generated but not finitely presented. Such tori have no minimal Seifert manifolds. An example of an embedded torus with the latter property was given recently by T. Maeda using different methods.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

scholarly journals Algebraic numbers, free group automorphisms and substitutions on the plane

2011 ◽  
Vol 363 (9) ◽  
pp. 4651-4699 ◽  
Author(s):  
Pierre Arnoux ◽  
Maki Furukado ◽  
Edmund Harriss ◽  
Shunji Ito
Keyword(s):  
Free Group ◽  
Algebraic Numbers ◽  
Group Automorphisms ◽  
Free Group Automorphisms

scholarly journals North–South dynamics of hyperbolic free group automorphisms on the space of currents

2019 ◽  
Vol 11 (02) ◽  
pp. 427-466 ◽  
Author(s):  
Martin Lustig ◽  
Caglar Uyanik
Keyword(s):  
Free Group ◽  
Outer Automorphism ◽  
Train Track ◽  
Geodesic Currents ◽  
Group Automorphisms ◽  
Free Group Automorphisms

Let [Formula: see text] be a hyperbolic outer automorphism of a non-abelian free group [Formula: see text] such that [Formula: see text] and [Formula: see text] admit absolute train track representatives. We prove that [Formula: see text] acts on the space of projectivized geodesic currents on [Formula: see text] with generalized uniform North-South dynamics.

scholarly journals Mapping Tori of Free Group Automorphisms are Coherent

1999 ◽  
Vol 149 (3) ◽  
pp. 1061 ◽  
Author(s):  
Mark Feighn ◽  
Michael Handel
Keyword(s):  
Free Group ◽  
Mapping Tori ◽  
Group Automorphisms ◽  
Free Group Automorphisms

scholarly journals Long turns, INP’s and indices for free group automorphisms

2015 ◽  
Vol 59 (4) ◽  
pp. 1087-1109 ◽  
Author(s):  
Thierry Coulbois ◽  
Martin Lustig
Keyword(s):  
Free Group ◽  
Group Automorphisms ◽  
Free Group Automorphisms

SET-THEORETIC YANG–BAXTER SOLUTIONS VIA FOX CALCULUS

2006 ◽  
Vol 15 (08) ◽  
pp. 949-956 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Baxter Equation ◽  
Group Representations ◽  
Group Automorphisms ◽  
Free Group Automorphisms

We construct solutions to the set–theoretic Yang–Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

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