Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Tagged geometric-group-theory — 5 entries. Clear
How large can the intersection of two free subgroups be? Hanna Neumann asked in 1957; Mineyev and Friedman proved the sharp bound independently in 2011.
Kaplansky asked whether torsion-free group rings have only trivial units. In 2021 Gardam found a counterexample over F₂ by computer search — the other conjectures survive.
If a hyperbolic group's boundary looks like a sphere, does the group act on hyperbolic 3-space? Cannon's 1990s conjecture, open in general.
One of the most famous yes/no questions in geometric group theory: does Thompson's group F admit an invariant mean? Open since the 1970s, resistant from both sides.
Are all hyperbolic groups residually finite? Gromov's basic question — no counterexample known, no proof in sight.
