Virtual FOG conjecture

Definitions and mathematical statement

Let $A$ be a primitive nonnegative integral matrix and $(X_A,\sigma)$ its mixing edge shift. Write $\operatorname{Aut}(X_A,\sigma)$ for the group of homeomorphisms commuting with $\sigma$. The ordered dimension group is $G_A=\varinjlim(\mathbb Z^N,v\mapsto vA)$, with its cone represented by nonnegative vectors and its automorphism $\delta_A[v,j]=[vA,j]$.

Each shift automorphism $\phi$ acts on this group, giving the dimension representation $\rho_A$. Concretely, an elementary strong shift equivalence $A=UV$, $B=VU$ induces $[v,j]\mapsto[vU,j]$; decomposing a conjugacy into these steps and a shift power defines its induced map, independently of the decomposition. Define the inert subgroup \[ \operatorname{Aut}_0(X_A)=\ker\rho_A, \qquad F_0(X_A)=\langle\phi\in\operatorname{Aut}_0(X_A): \phi^r=\operatorname{id}\text{ for some }r\ge1\rangle. \] The subgroup $F_0$ is normal, since conjugation preserves finite order and preserves the inert subgroup.

The virtual finite-order-generation conjecture asks whether \[ |\operatorname{Aut}_0(X_A)/F_0(X_A)|<\infty \] for every mixing SFT. The finite quotient may depend on $A$; there is no proposed uniform bound. Generation allows arbitrary finite products of finite-order inert automorphisms, without a common order bound. The stronger assertion $F_0=\operatorname{Aut}_0$ is not requested: the source records counterexamples to that earlier conjecture. Nor does the statement claim that every element of $F_0$ itself has finite order. It asks only for finite index of the subgroup generated by all such elements.

Short English statement

Do finite-order inert automorphisms generate a subgroup of finite index in the inert automorphism group of every mixing SFT?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600036 (AMR-045-0036), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 22.2. Original source indexed by AMR-045-0036. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.