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Looks like prunus laurocerasus, cherry laurel, an evergreen that can take drastic pruning. It has a lovely smell when it's cut as well so do it a little and often, it'll soon become a feature or a back drop to your garden.

I think it's your side so your responsibility, but their shrub/tree so offer the cut stuff back to them. Silly rules involving plants so try to resolve amicably....nothing worse than an angry neighbour http://www.problemneighbours.co.uk/rights-trees-and-overhanging-branches.html