Chain groups of homeomorphisms of the interval

hyun Kim, Sang; Koberda, Thomas; Lodha, Yash

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 4, pp. 797-820.

Nous introduisons la notion d'un groupe de chaînes d'homéomorphismes d'une variété de dimension un, ce qui est une certaine généralisation du groupe $F$ de Thompson. La classe des groupes qui en résulte profite de quelques phénomènes d'uniformité et de diversité. D'un côté, un groupe de chaînes possède un sous-groupe de commutateur simple, sinon l'action du groupe possède un intervalle d'errance. Dans ce dernier cas, le groupe de chaînes admet un quotient canonique qui est aussi un groupe de chaînes dont le sous-groupe de commutateur est simple. D'autre part, chaque sous-groupe engendré d'un sous-ensemble fini de ${Homeo}^{+}\left(I\right)$ peut être réalisé comme sous-groupe d'un groupe de chaînes. Il en résulte que les classes d'isomorphisme des groupes de chaînes sont indénombrables, ainsi que les classes d'isomorphisme des sous-groupes simples dénombrables de ${Homeo}^{+}\left(I\right)$ sont indénombrables. En outre, nous considérons les restrictions imposées sur les groupes de chaînes par la régularité, et nous démontrons l'existence de nombreux groupes de 3-chaînes qui n'admettent aucune action fidèle de classe $C^2$ sur une variété de dimension un, et de nombreux groupes de 6-chaînes qui n'admettent aucune action de classe $C^1$ sur une variété de dimension un. Il en résulte que les classes d'isomorphisme des sous-groupes simples dénombrables de ${Homeo}^{+}\left(I\right)$ qui n'agissent pas d'une manière non triviale sur l'intervalle sont indénombrables. Enfin, nous démontrons qu'un groupe de chaînes qui agit sur l'intervalle d'une manière minimale agit d'une manière unique, à conjugué topologique près.

We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson's group $F$. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. On the other hand, every finitely generated subgroup of ${Homeo}^{+}\left(I\right)$ can be realized as a subgroup of a chain group. As a corollary, we show that there are uncountably many isomorphism types of chain groups, as well as uncountably many isomorphism types of countable simple subgroups of ${Homeo}^{+}\left(I\right)$. We consider the restrictions on chain groups imposed by actions of various regularities, and show that there are uncountably many isomorphism types of 3-chain groups which cannot be realized by $C^2$ diffeomorphisms, as well as uncountably many isomorphism types of 6-chain groups which cannot be realized by $C^1$ diffeomorphisms. As a corollary, we obtain uncountably many isomorphism types of simple subgroups of ${Homeo}^{+}\left(I\right)$ which admit no nontrivial $C^1$ actions on the interval. Finally, we show that if a chain group acts minimally on the interval, then it does so uniquely up to topological conjugacy.

Classification : 20F60, 57M60, 20F65.
Keywords: homeomorphism, Thompson's group $F$, simple group, smoothing, chain group, orderable group, Rubin's Theorem
Mot clés : Homéomorphisme, groupe $F$ de Thompson, groupe simple, lissage, groupe de chaînes, groupe ordonnable, théorème de Rubin

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