Is there a ground between Set Theory and Group Theory/Algebra?

You may want to determine which discipline of set theory to link to another discipline. Descriptive Set Theory came (roughly) as a result of foundational issues arising from looking at certain arguments in Topology and Real Analysis, and at some point later ties to Model Theory, Proof Theory, and Recursion Theory were also investigated.

If you look at results in Universal Algebra, you will find many links to various foundational disciplines. This is probably the easiest source to find the kinds of links you mention. For example, as a weak parallel to Shelah's Classification Theory, one finds looking at varieties of algebras and considering their spectra, and classifying those which have many models in algebraic terms to those which have few models. People such as Baldwin, Jeong, Jezek, Kearnes, McKenzie, Valeriote, and Wood do work on decidability, the lattice of interpretability types, spectra, and other questions in the context of varieties.

There is also algebraic logic, cylindric algebras, and algebras being used to study certain aspects of set theory and logic. You may want to peruse some of that material and then revisit the question of what links you would still like to see.

Gerhard "Ask Me About System Design" Paseman, 2010.04.06