The goal of this paper is to investigate the physical nature of galaxies in the redshift range $0.02<z<0.15$ that have strong excess emission at mid-IR wavelengths and to determine whether they host a population of accreting black holes that cannot be identified using optical emission lines. We show that at fixed stellar mass $M_*$ and $D_n(4000)$, the distribution of [3.4]-[4.6] $mu$m (WISE W1-W2 band) colours is sharply peaked, with a long tail to much redder W1-W2 colours. We introduce a procedure to pull out the red outlier population based on a combination of three stellar population diagnostics. When compared with optically-selected AGN, red outliers are more likely to be found in massive galaxies, and they tend to have lower stellar mass densities, younger stellar ages and higher dust content than optically-selected AGN hosts. They are twice as likely to be detected at radio wavelengths. We examine W1-W2 colour profiles for a subset of the nearest, reddest outliers and find that most are not centrally peaked, indicating that the hot dust emission is spread throughout the galaxy. We find that radio luminosity is the quantity that is most predictive of a redder central W1-W2 colour. Radio-loud galaxies with centrally concentrated hot dust emission are almost always morphologically disturbed, with compact, unresolved emission at 1.4 Ghz. Eighty percent of such systems are identifiable as AGN using optical emission line diagnostics.
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