Translation:There are a lot of men.
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There is a subtle difference between 多い [おおい] and 沢山 [たくさん]。多い can suggest that there is more than necessary, an almost extensive amount, or more than there was previously. たくさん is 'a lot' in the general sense.
If someone told me that a library has 多くの本 vs 沢山の本, I'd say the first is overflowing with so many books!
ます indicates action, です indicates state of being, I believe. います indicates the action of existing in a certain place (I think, I'm not native). Using 多い with います would be incorrect in this context. You could use 多い in a different context, though (「ここに多い人います！」, "There are a lot of people here!"), but just be careful (「ここに多い人です！」, "Here is many people!").