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They are looking at the number of likes a page on your site has, such as a blog post that gets 17 likes on that particular URL. They're not reporting on the number of likes your page on FB has.

Hi Stu, as the root domain and sub-domain are targeting different geo-locations, moreoever Google treats the sub-domains a little more like a stand alone website, it is highly recommended to earn links for both of them individually, by all possible means from their respective geographies, I mean from the web properties operating or targeting audience in those geo-locations. For example, it would be odd for me to earn links for a Japanese website, targeting Japanese audience from web properties operating or targeting USA audience. Though this is not uncommon, it will not serve the purpose. Over and above this, fetching quality back links from any where is always recommended especially if those links are from authority web properties with high DA, PA and other SEO goodies like good Google PageRank etc.. I would conclude by addressing your question that fetching links to the root domain will not help much to make the sub-domain rank high in search results for many obvious reasons. Just like  we do deep link promotions for promoting pages other than the home page so that they can rank high for their respective target terms, we should be promoting the sub-domains also as if they were standalone websites so as to get maximum benefit from an off-page promotional campaign. Hope it helps my friend. Best, Devanur Rafi