Hi Ben 

Short answer -no.

Directly from Facebook Help:

It's currently not possible to advertise for a Group using Facebook ads. You can create other types of ads though for a Page. Facebook Blueprint is a great introduction to creating ads: https://www.facebook.com/blueprint/?ref=u2u

You can learn how to create an ad here: https://www.facebook.com/help/633662000000451/?ref=u2u

You can learn about the different placements you can choose from for your ads here: https://www.facebook.com/help/407108559393196/?ref=u2u

You can also get very specific with your targeting options: https://www.facebook.com/business/help/433385333434831/?ref=u2u

How much it costs to advertise depends entirely on the amount of money you'd like to spend. You can learn about the cost of advertising here: https://www.facebook.com/help/214319341922580/?ref=u2u

Additionally, here are some resources to help you get started: 
https://www.facebook.com/about/ads/?ref=u2u
https://www.facebook.com/advertising/?ref=u2u
https://www.facebook.com/business/products/ads/?ref=u2u
https://www.facebook.com/business/ads-guide/?ref=u2u
https://www.facebook.com/business/learn/?ref=u2u

You might also want to check out these 10 Answers to FAQS, voted most useful by the community! → http://on.fb.me/1VaVR6b

No, it's not. I do know people who use the strategy of creating a friendly URL that is used in the ad, that then redirects to the Facebook group.