Systolic invariants of groups and $2$-complexes via Grushko decomposition

Rudyak, Yuli B.; Sabourau, Stéphane

Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 777-800.

Nous prouvons un résultat de finitude pour l’aire systolique des groupes. Précisément, nous montrons qu’il n’existe qu’un nombre fini de facteurs non-libres dans les groupes fondamentaux des $2$-complexes d’aire systolique uniformément bornée. Nous montrons aussi que le nombre de tels groupes librement indécomposables croît au moins exponentiellement avec la borne sur l’aire systolique. De plus, nous prouvons une inégalité systolique uniforme pour tous les $2$-complexes de groupe fondamental non-libre qui améliore les bornes précédemment connues dans cette dimension.

We prove a finiteness result for the systolic area of groups. Namely, we show that there are only finitely many possible unfree factors of fundamental groups of $2$-complexes whose systolic area is uniformly bounded. We also show that the number of freely indecomposable such groups grows at least exponentially with the bound on the systolic area. Furthermore, we prove a uniform systolic inequality for all $2$-complexes with unfree fundamental group that improves the previously known bounds in this dimension.

Classification : 53C23, 20E06
Keywords: Systole, systolic area, systolic ratio, $2$-complex, Grushko decomposition
Mot clés : systole, aire systolique, rapport systolique, $2$-complexe, décomposition de Grushko

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