[Paper Review] Finitely presented groups associated with expanding maps

Volodymyr Nekrashevych|arXiv (Cornell University)|Dec 19, 2013

Homotopy and Cohomology in Algebraic Topology14 references18 citations

TL;DR

This paper introduces a finitely presented group $\mathcal{V}_f$ associated with every locally expanding self-covering map $f:\mathcal{M}\to\mathcal{M}$ on a compact, path-connected metric space. The group is constructed from the action of lifts of paths on the boundary of a rooted tree of preimages, and it serves as a complete topological invariant: $\mathcal{V}_{f_1} \cong \mathcal{V}_{f_2}$ if and only if $f_1$ and $f_2$ are topologically conjugate. The commutator subgroup is simple, and the abelianization has a dynamical interpretation.

ABSTRACT

We associate with every locally expanding self-covering $f:M o M$ of a compact path connected metric space a finitely presented group $V_f$. We prove that this group is a complete invariant of the dynamical system: two groups $V_{f_1}$ and $V_{f_2}$ are isomorphic as abstract groups if and only if the corresponding dynamical systems are topologically conjugate. We also show that the commutator subgroup of $V_f$ is simple, and give a topological interpretation of $V_f/V_f'$.

Motivation & Objective

• To establish a finitely presented group invariant for locally expanding self-covering maps on compact, path-connected metric spaces.
• To show that this group $\mathcal{V}_f$ is a complete invariant of the dynamical system: isomorphism of groups iff topological conjugacy of maps.
• To prove that the commutator subgroup of $\mathcal{V}_f$ is simple, and to give a topological interpretation of the abelianization $\mathcal{V}_f / \mathcal{V}_f'$.

Proposed method

• Construct the rooted tree $T_t$ of preimages of a basepoint $t \in \mathcal{M}$ under iterated $f$-pullbacks.
• Define homeomorphisms $S_\gamma$ on the boundary $\partial T_t$ induced by path lifts, which act as local isomorphisms on subtrees.
• Define $\mathcal{V}_f$ as the group of all homeomorphisms of $\partial T_t$ locally equal to such $S_\gamma$ maps.
• Show that $\mathcal{V}_f$ is generated by the Higman-Thompson group and the iterated monodromy group $\mathrm{IMG}(f)$, which captures monodromy data of $f$.
• Use groupoid-theoretic duality and hyperbolicity to relate the group $\mathcal{V}_f$ to the dynamics of $f$, especially via germs and natural extensions.
• Apply M. Rubin’s theorem on group actions on Cantor sets to reconstruct the dynamical system from $\mathcal{V}_f$.

Experimental results

Research questions

• RQ1 Can a finitely presented group be canonically associated with every locally expanding self-covering map on a compact, path-connected metric space?
• RQ2 Is the group $\mathcal{V}_f$ a complete invariant of the dynamical system, in the sense that isomorphism of groups implies topological conjugacy of maps?
• RQ3 What is the structure of the commutator subgroup $\mathcal{V}_f'$, and does it have special properties like simplicity?
• RQ4 How does the abelianization $\mathcal{V}_f / \mathcal{V}_f'$ relate to the dynamics of $f$?
• RQ5 Can the original dynamical system be reconstructed from the group $\mathcal{V}_f$ alone?

Key findings

• The group $\mathcal{V}_f$ is finitely presented for every locally expanding self-covering $f:\mathcal{M}\to\mathcal{M}$ on a compact, path-connected metric space.
• The group $\mathcal{V}_f$ is a complete invariant: $\mathcal{V}_{f_1} \cong \mathcal{V}_{f_2}$ as abstract groups if and only if $f_1$ and $f_2$ are topologically conjugate.
• The commutator subgroup $\mathcal{V}_f'$ is simple, as proven via general facts on self-similar groups and their associated groups.
• The abelianization $\mathcal{V}_f / \mathcal{V}_f'$ admits a topological interpretation as the fundamental group of the limit solenoid associated with $f$, or equivalently, the first homology of the limit space.
• The group $\mathcal{V}_f$ is isomorphic to $\mathcal{V}_{\mathrm{IMG}(f)}$, meaning it is fully determined by the iterated monodromy group of $f$, but without requiring the additional self-similar structure.
• The dynamical system $f$ can be reconstructed from $\mathcal{V}_f$ via the groupoid of germs and the action on the Cantor set $\partial T_t$, establishing a complete classification.