Andrews–Curtis conjecture for free groups

Prove that the AC-graph \(\Delta_k^S(F_k)\), whose vertices are the \(k\)-element normal generating sets of the free group \(F_k\) and whose edges are generated by the general Andrews–Curtis transformations associated with a generating set \(S\), is connected.

Background

The paper reformulates the Andrews–Curtis problem as a graph-navigation problem. For a group GG, let Nk(G)N_k(G) denote the kk-tuples that normally generate GG, and let GACk(G)GAC_k(G) be the group of Andrews–Curtis transformations preserving normal generation. A finite generating set SACk(G,S)SAC_k(G,S) induces the AC-graph ΔkS(G)\Delta_k^S(G) on Nk(G)N_k(G).

For the free group FkF_k, the conjecture asserts that this graph is connected, equivalently that every normal generating kk-tuple can be transformed into a canonical one by Andrews–Curtis moves. The paper notes that finite-group analogues are known, while the free-group case remains the unresolved problem identified here.

References

The Andrews-Curtis conjecture is that \DeltaS_k(F_k) is connected (one can take S to be any set of generators).

— Diffusion Models for Cayley Graphs  (2503.05558 - Douglas et al., 7 Mar 2025) in Section 2, item “The Andrews-Curtis problem” in the discussion of Cayley graphs of group actions

Adjoining the relation u=1 to this presentation is AC-equivalent to \langle x_0, x_1, x_2: u, \theta(u), \theta2(u) \rangle, where it is an open problem to determine if this presentation of the trivial group is AC-equivalent to \langle x_0, x_1, x_2: x_0, x_1, x_2 \rangle (this is referred to as Neumann's potential counterexample in the literature ).

— On $\textbf{u}$-substitutions for group presentations  (2608.18929 - Mcdermott, 19 Aug 2026) in Section 3, subsection “Groups of weight one”