We say 'There are many people' and 'There is a group of people'. (In spoken English, 'There's three people.' is OK.)

What about 'There _____ a lot of people'. 'Is' or 'are', and why?

I consulted Fowler's Modern English Usage (2nd Ed., revised by Gowers) which has this to say:

Under he heading lot: "A l. of people say so", "Lots of paper is wanted", etc. See Number 6(b)."

Under the heading Number 6(b) we find:

quote:

"There are heaps more to say, but I must not tax your space further." The plurals heaps and lots used colloquially for a great amount now always take a singular verb unless a plural noun with of is added: "There is heaps of ammunition", but "There are heaps of cups"; "There is lots to do", but "Lots of people think so."

Thus, acording to Fowler's, your missing word is are.

As for the undisputed authority of this particular reference book, there was an article in The New Yorker November 26, 2001, by Louis Menand lauding it as "one of the fruits of the Oxford English Dictionary." Actually the article was about Fowler, not Fowler's, but never mind that. Menand adds that "Language is a social weapon. Fowler was mostly trying to keep it from blowing up in peoples' hands."