It is well known that the group of diffeomorphisms of the circle contains free non-Abelian subgroups. Is it true (known) that the group of diffeomorphisms of the interval $[0,1]$ contains free subgroups? One approach to get a positive answer can be the following. Consider all functions $f_a=\frac{\exp(ax)-1}{\exp(a)-1}$, $a>1$, which are smooth diffeomorphisms of $[0,1]$. First prove that the group generated by these functions does not satisfy any non-trivial law. Now for any non-trivial word $w$ in two variables consider the function $w(f_a,f_b)$ (for every $a,b$). The set of pairs $(a,b)\in \mathbb{R}^2$ for which this function is the identity function has dimension at most 1. Then by Baire category theorem, since the set of words $w$ is countable, there are $a,b$ such that $f_a,f_b$ freely generate a free subgroup. Does this argument actually work?
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$\begingroup$ This kind of question (zoology of subgroups of diffeomorphisms of intervals) it can be important to mention the differentiability class ($Diff^\infty$, $Diff^2$, $Diff^1$). Examples by Navas show that they are very different. I assume you mean $Diff^\infty$? $\endgroup$– YCorSep 20, 2012 at 7:24
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$\begingroup$ @Yves Cornulier : The reference in my answer shows that the differentiability class does not matter for this particular question. $\endgroup$– Andy PutmanSep 21, 2012 at 3:16
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Much more is true. The compactly supported diffeomorphism group of any (positive-dimensional, nonempty) manifold contains free subgroups of uncountable rank. In fact, there are such subgroups that are generated by sets which are arcwise connected! See the paper
MR0974661 (90b:58031) Grabowski, Janusz(PL-WASW) Free subgroups of diffeomorphism groups. Fund. Math. 131 (1988), no. 2, 103–121.
which is available online here.
answered Sep 20, 2012 at 6:26
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4$\begingroup$ "..subgroups that are generated by sets which are arcwise connected!" Wow, that is an amazing concept! $\endgroup$– David Roberts ♦Sep 20, 2012 at 6:34