Soit le groupe libre engendré par n éléments, et soit le groupe des automorphismes de qui envoient chaque générateur sur un conjugué. Le noyau de l'homomorphisme , obtenu en envoyant un des générateurs du groupe libre sur l'identité, est de type fini. On démontre que est de dimension cohomologique , est que n'est pas de type fini pour . Par conséquent n'est pas de présentation finie pour .
Let be the free group on n generators, and let be the group of automorphisms of that send each generator to a conjugate of itself. The kernel of the homomorphism , induced by mapping one of the free group generators to the identity, is finitely generated. We show that has cohomological dimension , and that is not finitely generated for . It follows that is not finitely presentable for .
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@article{CRMATH_2010__348_3-4_127_0, author = {Pettet, Alexandra}, title = {Finiteness properties for a subgroup of the pure symmetric automorphism group}, journal = {Comptes Rendus. Math\'ematique}, pages = {127--130}, publisher = {Elsevier}, volume = {348}, number = {3-4}, year = {2010}, doi = {10.1016/j.crma.2009.12.011}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2009.12.011/} }
TY - JOUR AU - Pettet, Alexandra TI - Finiteness properties for a subgroup of the pure symmetric automorphism group JO - Comptes Rendus. Mathématique PY - 2010 SP - 127 EP - 130 VL - 348 IS - 3-4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2009.12.011/ DO - 10.1016/j.crma.2009.12.011 LA - en ID - CRMATH_2010__348_3-4_127_0 ER -
%0 Journal Article %A Pettet, Alexandra %T Finiteness properties for a subgroup of the pure symmetric automorphism group %J Comptes Rendus. Mathématique %D 2010 %P 127-130 %V 348 %N 3-4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2009.12.011/ %R 10.1016/j.crma.2009.12.011 %G en %F CRMATH_2010__348_3-4_127_0
Pettet, Alexandra. Finiteness properties for a subgroup of the pure symmetric automorphism group. Comptes Rendus. Mathématique, Tome 348 (2010) no. 3-4, pp. 127-130. doi : 10.1016/j.crma.2009.12.011. http://archive.numdam.org/articles/10.1016/j.crma.2009.12.011/
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