Is it known whether there is any example of a pair of rotations in $SO(3)$ about orthogonal axes such that the group that they generate is not a free product of the two cyclic groups generated by each generator?
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2$\begingroup$ What about two rotations of angle $\pi $? They commute, so they certainly do not span a free product. $\endgroup$– abxCommented Nov 27, 2014 at 11:00
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3$\begingroup$ Yes, the group of symmetries of the cube is an example. Did you mean to ask something else? $\endgroup$– Robert BryantCommented Nov 27, 2014 at 11:00
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Yes, there are many infinite examples in a paper of Radin and Sadun, some of which were rediscovered in a paper of mine via different methods. For example, the group $$\langle \left( \begin{array}{clcr} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 &0& 1 \end{array} \right), \left( \begin{array}{clcr} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ 0& \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{array} \right) \rangle $$ is isomorphic to the amalgam $S_{4}{\ast}_{D_{8}}{D_{16}},$ where I am using $D_{m}$ to denote the dihedral group with $m$ elements.
Note that the Euler characteristic ( as extended by Wall) of this group is $\frac{-1}{48},$ whereas the Euler characteristic of the free product $C_{4} {\ast} C_{8}$ is $\frac{-5}{8}.$
answered Nov 27, 2014 at 11:10
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$\begingroup$ Thank you, what about examples where one of the generators has infinite order? $\endgroup$– RupertCommented Nov 27, 2014 at 11:44
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$\begingroup$ I haven't looked at that myself. $\endgroup$ Commented Nov 27, 2014 at 11:51