On the intersection of subgroups of a free group

1992 ◽  
Vol 108 (1) ◽  
pp. 29-36 ◽  
Author(s):  
Gábor Tardos
Keyword(s):  
Free Group ◽  
Intersection Of Subgroups

scholarly journals INTERSECTION OF SUBGROUPS IN FREE GROUPS AND HOMOTOPY GROUPS

2008 ◽  
Vol 18 (05) ◽  
pp. 803-823 ◽  
Author(s):  
HANS-JOACHIM BAUES ◽  
ROMAN MIKHAILOV
Keyword(s):  
Abelian Group ◽  
Free Group ◽  
Homotopy Group ◽  
Homotopy Groups ◽  
Natural Homomorphism ◽  
Two Dimensional ◽  
The Third ◽  
Cw Complex ◽  
Intersection Of Subgroups ◽  
The Right

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group π3. This generalizes a result of Gutierrez–Ratcliffe who relate the intersection of two subgroups with the computation of π2. Let K be a two-dimensional CW-complex with subcomplexes K1, K2, K3 such that K = K1 ∪ K2 ∪ K3 and K1 ∩ K2 ∩ K3 is the 1-skeleton K1 of K. We construct a natural homomorphism of π1(K)-modules [Formula: see text] where Ri = ker {π1(K1) → π1(Ki)}, i = 1,2,3 and the action of π1(K) = F/R1R2R3 on the right-hand abelian group is defined via conjugation in F. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

scholarly journals The minimally displaced set of an irreducible automorphism of $$F_N$$ is co-compact

2021 ◽  
Vol 116 (4) ◽  
pp. 369-383
Author(s):  
Stefano Francaviglia ◽  
Armando Martino ◽  
Dionysios Syrigos
Keyword(s):  
Growth Rate ◽  
Free Group ◽  
Exponential Growth ◽  
Single Point ◽  
Irreducible Automorphism

AbstractWe study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $$\phi $$ ϕ , under the action of the centraliser $$C(\phi )$$ C ( ϕ ) . As a corollary, we get that the same holds for the action of $$ <\phi>$$ < ϕ > on $$Min(\phi )$$ M i n ( ϕ ) . Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one consists of a single point.

Subgroups induced by certain ideals of free group rings

1983 ◽  
Vol 11 (22) ◽  
pp. 2519-2525 ◽  
Author(s):  
Chander Kanta Gupta
Keyword(s):  
Free Group ◽  
Group Rings

scholarly journals Graphical splittings of Artin kernels

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrique Miguel Barquinero ◽  
Lorenzo Ruffoni ◽  
Kaidi Ye
Keyword(s):  
Free Group ◽  
Artin Group ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Block Graphs ◽  
Right Angled Artin Groups

Abstract We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

scholarly journals The dynamics and geometry of free group endomorphisms

2021 ◽  
Vol 384 ◽  
pp. 107714
Author(s):  
Jean Pierre Mutanguha
Keyword(s):  
Free Group

On Embeddings of the Free Group into the Group of Infinite Unitriangular Matrices

2021 ◽  
Vol 252 (6) ◽  
pp. 782-783
Author(s):  
A. I. Generalov ◽  
A. S. Mironov
Keyword(s):  
Free Group

scholarly journals Comparison of ocular surface assessment and adherence between preserved and preservative-free latanoprost in glaucoma: a parallel-grouped randomized trial

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Dai Woo Kim ◽  
Jonghoon Shin ◽  
Chang Kyu Lee ◽  
Myungjin Kim ◽  
Sohyeon Lee ◽  
...  
Keyword(s):  
Free Group ◽  
Ocular Surface ◽  
Open Angle Glaucoma ◽  
Drug Tolerance ◽  
Adherence Rate ◽  
Eye Drops ◽  
Ocular Surface Disease Index ◽  
Significant Difference ◽  
Surface Assessment ◽  
Preservative Free

AbstractGiven that nonadherence is related to subject characteristics and drug tolerance and preserved eye drops tend to be more intolerable than preservative-free ones, we conducted a phase 4, parallel-grouped, investigator-blind, active-control, randomized, multicenter study. A total of 51 patients with intraocular pressure (IOP) ≥ 15 mmHg diagnosed with open-angle glaucoma or ocular hypertension were randomly assigned to the preserved latanoprost group (n = 26) and the preservative-free latanoprost group (n = 25). The efficacy variables were corneal/conjunctival staining grade, Ocular Surface Disease Index (OSDI), adherence at 12 weeks after the first administration; corneal/conjunctival staining grade at 4 weeks; and IOP, tear break-up time (TBUT), and hyperemia score at 4 and 12 weeks. The safety variables included visual acuity and drug tolerance questionnaire results. There was no statistically significant difference in corneal/conjunctival staining grade, OSDI, or TBUT between the groups at 4 and 12 weeks. However, the adherence rate was higher and the hyperemia score was lower in the preservative-free group than in the preserved group. The severity and duration of stinging/burning sensation were lower in the preservative-free group than in the preserved group. Overall, preservative-free latanoprost showed better ocular tolerance assessed by hyperemia scores and stinging/burning symptoms following higher adherence than preserved latanoprost.

scholarly journals The free splitting complex of a free group, II: Loxodromic outer automorphisms

2019 ◽  
Vol 372 (6) ◽  
pp. 4053-4105 ◽  
Author(s):  
Michael Handel ◽  
Lee Mosher
Keyword(s):  
Free Group ◽  
Outer Automorphisms ◽  
Group Ii
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