  Astrophysical black hole candidates are thought to be the Kerr black hole
predicted by General Relativity. In order to confirm the Kerr-nature of these
objects, we need to probe the geometry of the space-time around them and see if
the observations are consistent with the predictions of the Kerr metric. That
can be achieved, for instance, by studying the properties of the
electromagnetic radiation emitted by the gas in the accretion disk. The
high-frequency quasi-periodic oscillations observed in the X-ray flux of some
stellar-mass black hole candidates might do the job. As the frequencies of
these oscillations depend only very weakly on the observed X-ray flux, it is
thought they are mainly determined by the metric of the space-time. In this
paper, I consider the resonance models proposed by Abramowicz and Kluzniak and
I extend previous results to the case of non-Kerr space-times. The emerging
picture is more complicated than the one around a Kerr black hole and there is
a larger number of possible combinations between different modes. I then
compare the bounds inferred from the twin peak high-frequency quasi-periodic
oscillations observed in three micro-quasars (GRO J1655-40, XTE J1550-564, and
GRS 1915+105) with the measurements from the continuum-fitting method of the
same objects. For Kerr black holes, the two approaches do not provide
consistent results. In a non-Kerr geometry, this conflict may be solved if the
observed quasi-periodic oscillations are produced by the resonance $\nu_\theta
: \nu_r = 3:1$, where $\nu_\theta$ and $\nu_r$ are the two epicyclic
frequencies. It is at least worth mentioning that the deformation from the Kerr
solution required by observations would be consistent with the one suggested in
another recent work discussing the possibility that steady jets are powered by
the spin of these compact objects.
