How do sedges adapt in the tundra?

It depends on the type of "sedge" that you are referring to. Here are the different plants that can be referred to as "sedge" and where they live, as well as whether their range includes tundra.

Acoros calamus, also called Sweet Flag, lives in India (where it is native), Europe, southern Russia, northern Turkey, Japan, Burma, Sri Lanka, Australia, and China. It doesn't live in the tundra, although it does live in subarctic pine forests.

Cyperaceae, a group of flowering plants that live in tropical Asia and South America. They do not inhabit the tundra.

Iris pseudacoros, or Yellow Flag. I do not have specific information on where it lives, although it doesn't appear to live in the tundra.

You can see the related links for more information on each of these plants.