We present a measurement of the anisotropic void-galaxy cross-correlation function in the CMASS galaxy sample of the BOSS DR12 data release. We perform a joint fit to the data for redshift space distortions (RSD) due to galaxy peculiar velocities and anisotropies due to the Alcock-Paczynski (AP) effect, for the first time using a velocity field reconstruction technique to remove the complicating effects of RSD in the void centre positions themselves. Fits to the void-galaxy function give a 1% measurement of the AP parameter combination $D_A(z)H(z)/c = 0.4367pm 0.0045$ at redshift $z=0.57$, where $D_A$ is the angular diameter distance and $H$ the Hubble parameter, exceeding the precision obtainable from baryon acoustic oscillations (BAO) by a factor of ~3.5 and free of systematic errors. From voids alone we also obtain a 10% measure of the growth rate, $fsigma_8(z=0.57)=0.501pm0.051$. The parameter degeneracies are orthogonal to those obtained from galaxy clustering. Combining void information with that from BAO and galaxy RSD in the same CMASS sample, we measure $D_A(0.57)/r_s=9.383pm 0.077$ (at 0.8% precision), $H(0.57)r_s=(14.05pm 0.14);10^3$ kms$^{-1}$Mpc$^{-1}$ (1%) and $fsigma_8=0.453pm0.022$ (4.9%), consistent with cosmic microwave background (CMB) measurements from Planck. These represent a factor sim2 improvement in precision over previous results through the inclusion of void information. Fitting a flat cosmological constant $Lambda$CDM model to these results in combination with Planck CMB data, we find up to an 11% reduction in uncertainties on $H_0$ and $Omega_m$ compared to use of the corresponding BOSS consensus values. Constraints on extended models with non-flat geometry and a dark energy of state that differs from $w=-1$ show an even greater improvement.
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