Free products and the algebraic structure of diffeomorphism groups

Abstract

Let $M$ be a compact one--manifold, and let ${\mathrm{Diff}}^{1+\mathrm{bv}}(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation. We prove that if $G$ is a group which is not virtually metabelian, then $(G\times \mathbb{Z})\ast \mathbb{Z}$ is not realized as a subgroup of ${\mathrm{Diff}}^{1+\mathrm{bv}}(M)$. This gives the first examples of finitely generated groups $G,H\le {\mathrm{Diff}}_{+}^{\infty }(M)$ such that $G\ast H$ does not embed into ${\mathrm{Diff}}^{1+\mathrm{bv}}(M)$. By contrast, for all countable groups $G,H\le {\mathrm{Homeo}}^{+}(M)$ there exists an embedding $G\ast H\to {\mathrm{Homeo}}^{+}(M)$. We deduce that many common groups of homeomorphisms do not embed into ${\mathrm{Diff}}^{1+\mathrm{bv}}(M)$, for example the free product of $\mathbb{Z}$ with Thompson's group $F$. We also complete the classification of right-angled Artin groups which can act smoothly on $M$ and in particular, recover the main result of a joint work of the authors with Baik. Namely, a right-angled Artin group $A(\Gamma )$ either admits a faithful $C^{\infty }$ action on $M$, or $A(\Gamma )$ admits no faithful $C^{1+\mathrm{bv}}$ action on $M$. In the former case, $A(\Gamma )\cong {\prod }_i{G}_i$ where $G_i$ is a free product of free abelian groups. Finally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.

28 pages. To appear in the Journal of Topology

References in corpus (4)

• Spaces of invariant circular orders of groups
• Rigidity and flexibility of group actions on the circle
• On the sharp regularity for arbitrary actions of nilpotent groups on the interval: the case of N_4
• Diffeomorphism groups of critical regularity

Cited by in corpus (2)

• Rigidity of mapping class group actions on $S^1$
• Small $C^1$ actions of semidirect products on compact manifolds