Embedding surface groups in Thompson’s group V

Determine whether a closed higher-genus surface group embeds in Thompson’s group V.

Background

Thompson’s group T is a subgroup of Thompson’s group V, and the paper studies subgroup-embedding questions for these groups. The authors resolve the intermediary question of whether surface groups embed in T by proving that they do not. However, the broader question concerning embeddings of closed higher-genus surface groups into V remains unresolved and is explicitly identified as an open problem.