Geodesic
Some properties of $E(G,W,\mathcal{F}_TG)$ and~an~application in the theory of splittings of~groups
Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 179-193.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let us consider $W$ a $G$-set and $M$ a $\mathbb{Z}_2G$-module, where $G$ is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant $E(G,W,M)$, defined in [5] and present related results with independence of $E(G,W,M)$ with respect to the set of $G$-orbit representatives in $W$ and properties of the invariant $E(G,W,\mathcal{F}_TG)$ establishing a relation with the end of pairs of groups $\widetilde{e}(G,T)$, defined by Kropphller and Holler in [15]. The main results give necessary conditions for $G$ to split over a subgroup $T$, in the cases where $M=\mathbb{Z}_2(G/T)$ or $M=\mathcal{F}_TG$.
Keywords: cohomology of groups, cohomological invariants, splittings and derivation of groups. 
@article{ADM_2020_30_2_a2,
     author = {E. L. C. Fanti and L. S. Silva},
     title = {Some properties of $E(G,W,\mathcal{F}_TG)$ and~an~application in the theory of splittings of~groups},
     journal = {Algebra and discrete mathematics},
     pages = {179--193},
     publisher = {mathdoc},
     volume = {30},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a2/}
}
TY  - JOUR
AU  - E. L. C. Fanti
AU  - L. S. Silva
TI  - Some properties of $E(G,W,\mathcal{F}_TG)$ and~an~application in the theory of splittings of~groups
JO  - Algebra and discrete mathematics
PY  - 2020
SP  - 179
EP  - 193
VL  - 30
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a2/
LA  - en
ID  - ADM_2020_30_2_a2
ER  - 
%0 Journal Article
%A E. L. C. Fanti
%A L. S. Silva
%T Some properties of $E(G,W,\mathcal{F}_TG)$ and~an~application in the theory of splittings of~groups
%J Algebra and discrete mathematics
%D 2020
%P 179-193
%V 30
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a2/
%G en
%F ADM_2020_30_2_a2
E. L. C. Fanti; L. S. Silva. Some properties of $E(G,W,\mathcal{F}_TG)$ and~an~application in the theory of splittings of~groups. Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 179-193. http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a2/

[1] Andrade M. G. C., Invariantes relativos e dualidade, Tese (Doutorado em Matemática), IMECC-UNICAMP, Campinas, 1992

[2] Andrade M. G. C., Fanti E. L. C., “A relative cohomological invariant for pairs of groups”, Manuscripta Math., 83 (1994), 1–18 | DOI | MR | Zbl

[3] Andrade M. G. C., Fanti E. L. C., “A remark about amalgamation of groups and index of certain subgroups”, International Journal of Pure and Applied Mathematics, 23 (2010), 55–62 | MR | Zbl

[4] Andrade M. G. C., Fanti E. L. C., “A note about splittings of groups and commensurability under a cohomological point of view”, Algebra Discrete Math., 9:2 (2010), 1–10 | MR

[5] Andrade M. G. C., Fanti E. L. C., “The cohomological invariant $E'(G,W)$ and some properties”, International Journal of Applied Mathematics, 25:2 (2012), 183–190 | MR | Zbl

[6] Andrade M. G. C., Fanti E. L. C., Daccach J. A., “On certain relative cohomological invariant”, International Journal of Pure and Applied Mathematics (Bulgária), 21:3 (2005), 335–351 | MR | Zbl

[7] Andrade M. G. C., Fanti E. L. C., Fêmina L. L., “On Poincaré duality for pairs $(G,W)$”, Open Mathematics, 13 (2015), 363–371 | DOI | MR | Zbl

[8] Bieri R., Homological dimension of discrete groups, Queen Mary College Math. Notes, Londres, 1976 | MR | Zbl

[9] Bieri R., Eckmann B., “Relative homology and Poincaré duality for Group Pairs”, Journal of Pure and Applied Algebra, 13 (1978), 277–319 | DOI | MR | Zbl

[10] Brown K. S., Cohomology of groups, Springer-Verlag, New York, 1982 | MR | Zbl

[11] Dicks W., Dunwoody M., Groups acting on graphs, Cambridge University Press, Cambridge, 1988 | MR

[12] Freudenthal H., “Über die Enden topologischer Raüme und Gruppen”, Math. Zeit, 33 (1931), 692–713 | DOI | MR

[13] Hopf H., “Enden offener Raüme und unendliche diskontinuierliche Gruppen”, Math. Helv., 16 (1943), 81–100 | DOI | MR

[14] Houghton C. H., “Ends of locally compact groups and their coset spaces”, J. Aust. Math. Soc., 17 (1974), 274–284 | DOI | MR | Zbl

[15] Kropholler P. H., Roller M. A., “Relative ends and duality groups”, Journal off Pure and Appl. Algebra, 61 (1989), 197–210 | DOI | MR | Zbl

[16] Scott G. P., “Ends of pairs of groups”, J. Pure Appl. Algebra, 11 (1977), 179–198 | DOI | MR

[17] Scott G. P., Wall C. T. C., “Topological methods in group theory”, Homological Group Theory, London Math. Soc. Lect. Notes Series, 36, 1979, 137–203 | MR | Zbl

[18] Specker E., “Endenverbande von Raumen und Gruppen”, Math. Ann., 122 (1950), 167–174 | DOI | MR

[19] Stallings J. R., “On torsion-free groups with infinitely many ends”, Ann. Math., 1968, 312–334 | DOI | MR | Zbl