Submillimetre-selected galaxies (SMGs) at high redshift ($z$ $sim$ 2) are potential host galaxies of active galactic nuclei (AGN). If the local Universe is a good guide, $sim$ 50$%$ of the obscured AGN amongst the SMG population could be missed even in the deepest X-ray surveys. Radio observations are insensitive to obscuration; therefore, very long baseline interferometry (VLBI) can be used as a tool to identify AGN in obscured systems. A well-established upper limit to the brightness temperature of 10$^5$ K exists in star-forming systems, thus VLBI observations can distinguish AGN from star-forming systems via brightness temperature measurements. We present 1.6 GHz European VLBI Network (EVN) observations of four SMGs (with measured redshifts) to search for evidence of compact radio components associated with AGN cores. For two of the sources, e-MERLIN images are also presented. Out of the four SMGs observed, we detect one source, J123555.14, that has an integrated EVN flux density of 201 $pm$ 15.2 $mu$Jy, corresponding to a brightness temperature of 5.2 $pm$ 0.7 $times$ 10$^5$ K. We therefore identify that the radio emission from J123555.14 is associated with an AGN. We do not detect compact radio emission from a possible AGN in the remaining sources (J123600.10, J131225.73, and J163650.43). In the case of J131225.73, this is particularly surprising, and the data suggest that this may be an extended, jet-dominated AGN that is resolved by VLBI. Since the morphology of the faint radio source population is still largely unknown at these scales, it is possible that with a $sim$ 10 mas resolution, VLBI misses (or resolves) many radio AGN extended on kiloparsec scales.
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