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Abstract  Let   M   be a compact one‐manifold, and letDiff  +   1  +   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  ×  Z  )  ∗  Z    is not realized as a subgroup ofDiff  +   1  +   bv(  M  )    . This gives the first examples of finitely generated groups    G  ,  H  ⩽   Diff  +  ∞    (  M  )such that   G*H   does not embed intoDiff  +   1  +   bv(  M  )    . By contrast, for all countable groups    G  ,  H  ⩽   Homeo  +    (  M  )there exists an embedding    G  *  H  →   Homeo  +    (  M  )    . We deduce that many common groups of homeomorphisms do not embed intoDiff  +   1  +   bv(  M  )    , for example the free product of   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 [3][H. Baik, 2016]. Namely, a right‐angled Artin group    A  (  Γ  )    either admits a faithful    C  ∞    action on   M  , or    A  (  Γ  )    admits no faithful    C   1  +   bvaction on   M  . In the former case,A   (  Γ  )   ≅   ∏  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   .

Free products and the algebraic structure of diffeomorphism groups