Groups are the mathematical objects that capture the structure of symmetry.  One modern approach to the study of infinite groups is to apply combinatorial and geometric tools.  For a flavour of these techniques, you can scan the book “Office Hours with a Geometric Group Theorist”, edited by Matt Clay and Dan Margalit, 2017 (ISBN 9780691158662).  Projects may be expository or include original research.  Specific topics may include: groups and their graphs; the geometry of automorphism groups of groups; groups, algorithms and machines.