# Unbounded quasimorphisms and free subgroups

- Reference: `kourovka-21-48-unbounded-quasimorphisms-and-free-subgroups`
- Page: https://theoremdb.org/statements/kourovka-21-48-unbounded-quasimorphisms-and-free-subgroups
- Record maturity: Reviewed problem

## Problem

A quasimorphism on a group G is a function f: G → R such that the quantity sup |f (g) + f (h) − f (gh)| is finite. A quasimorphism is homogeneous if it restricts to g,h a homomorphism on every cyclic subgroup of G. Let G be a group admitting an unbounded homogeneous quasimorphism G → R that is not a homomorphism. Must G contain a non-abelian free subgroup? This is Kourovka Notebook Problem \(21.48\).

## Status

The reviewed record remains open.

## Work

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `kourovka-21-48-unbounded-quasimorphisms-and-free-subgroups`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.