Cayley-tree pseudo-orbit tracing: period subgroups and relative geometry

Abstract: We introduce Cayley-tree POTP, obtained by imposing the pseudo-orbit equations of a finitely generated group action only along a spanning tree of a Cayley graph. For zero-dimensional actions, we characterize this property by equicontinuity along normalized replacement paths; for subshifts, the criterion is expressed in the right-coset space of the common left-period subgroup. These criteria characterize virtual freeness and, for commensurated subgroup pairs, identifies Cayley-tree POTP of the coset full shift with relative quasi-tree geometry and a finite Bass--Serre decomposition. For infinite-index VFP pairs, Cayley-tree POTP of the coset full shift is equivalent to virtual cohomological codimension one, although ordinary POTP holds for every such shift. Finally, we prove that the strong topological Rokhlin property passes to finite-index overgroups. Consequently every finitely generated virtually free group has this property, answering the virtually cyclic case posed by Doucha, and Cayley-tree POTP is generic for its Cantor actions.

• The paper establishes a replacement-path criterion showing that Cayley-tree pseudo-orbit tracing combines ordinary shadowing with equicontinuity, or finite coset conditions for subshifts.
• The results show that full shifts have Cayley-tree tracing exactly for virtually free groups, while commensurated coset shifts correspond to quasi-tree orbital geometry and finite graph-of-groups structures.
• The paper connects Cayley-tree tracing for VFP groups to virtual cohomological codimension one and proves its genericity for Cantor actions of finitely generated virtually free groups via STRP permanence.

The Cayley-tree pseudo-orbit tracing property

This paper, "Cayley-tree pseudo-orbit tracing: period subgroups and relative geometry" (2608.16483), introduces a refinement of the pseudo-orbit tracing property (POTP, also called shadowing) for actions of finitely generated groups. Given a finite symmetric generating set $S=S^{-1}$ of $\Gamma$, a $\delta$-pseudo-orbit is a family $(x_g)_{g\in\Gamma}$ in a compact $\Gamma$-space satisfying $d(ax_g,x_{ag})<\delta$ for all Cayley edges. The paper weakens this requirement by imposing the inequalities only along a spanning tree $T\subseteq C_S$: an $(H,\delta)$-pseudo-orbit for a spanning subgraph $H$ satisfies the inequalities only on edges of $H$. The action has $\Gamma$0-POTP if every such pseudo-orbit is $\Gamma$1-traced for suitable $\Gamma$2, and it has Cayley-tree POTP if $\Gamma$3 is a spanning tree for some choice of $\Gamma$4 and $\Gamma$5. Since deleting edges weakens the constraints, $\Gamma$6-POTP always implies ordinary POTP; the substantive question is the converse, and the paper shows the converse holds for all actions exactly when $\Gamma$7 is virtually free.

The notion is distinct from tree-shifts indexed by rooted trees and from translation-invariant support formalisms, because a Cayley spanning tree may retain different generator edges at different vertices.

The replacement-path criterion

The central technical device is the normalized replacement-path set

$\Gamma$8

where $\Gamma$9 is the unique simple path in $\delta$0. This set generates $\delta$1 but is not generally a subgroup. The main criterion states:

• For a zero-dimensional compact metrizable action, $\delta$2-POTP holds if and only if the action has ordinary POTP and the family $\delta$3 is equicontinuous.
• For a nonempty subshift $\delta$4, $\delta$5-POTP holds if and only if $\delta$6 is a subshift of finite type (SFT) and $\delta$7 for every $\delta$8, where $\delta$9 is the common left-period subgroup $(x_g)_{g\in\Gamma}$0. Equivalently, the restricted shifts $(x_g)_{g\in\Gamma}$1 are equicontinuous.

The proof of the converse direction uses a clopen-partition argument: equicontinuity forces transported values along a replacement path into a single atom of the partition, upgrading a tree pseudo-orbit to an ordinary pseudo-orbit. A key lemma shows that the finite-determinacy sets $(x_g)_{g\in\Gamma}$2 are exactly finite unions of right $(x_g)_{g\in\Gamma}$3-cosets, connecting equicontinuity to the coset-finiteness condition. Notably, $(x_g)_{g\in\Gamma}$4 need not be normal; the action kernel is its normal core.

Full shifts, bounded stretch, and virtual freeness

For the full shift, $(x_g)_{g\in\Gamma}$5, so the criterion reduces to bounded stretch $(x_g)_{g\in\Gamma}$6, i.e., $(x_g)_{g\in\Gamma}$7 being a tree spanner. The main structural theorem gives four equivalent conditions for $(x_g)_{g\in\Gamma}$8: (i) $(x_g)_{g\in\Gamma}$9 is virtually free; (ii) some Cayley graph admits a spanning tree with finite stretch; (iii) the full shift $\Gamma$0 has Cayley-tree POTP; (iv) every compact action with ordinary POTP has Cayley-tree POTP. The geometric equivalence (i)–(ii) is Antolín's theorem on abstract uniform trees; the dynamical content is that Cayley-tree POTP of a single full shift detects virtual freeness. The proof enlarges the generating set by a finite ball so that an abstract uniform tree becomes an actual Cayley spanning tree.

Two examples sharpen the picture. The full shift over $\Gamma$1 has ordinary POTP (being an SFT) but no Cayley-tree POTP for any tree in any Cayley graph. And for $\Gamma$2 with the ladder Cayley graph, two spanning trees—one rail plus all rungs versus both rails joined by a single rung—give $\Gamma$3-POTP but not $\Gamma$4-POTP for the same action, so the property genuinely depends on the tree.

Period subgroups: finite height and commensuration

For coset full shifts $\Gamma$5, the left-period subgroup equals $\Gamma$6, and the shift is an SFT exactly when $\Gamma$7 is finitely generated. The coset-finiteness condition $\Gamma$8 behaves differently in two regimes.

Finite-intersection regime. If finitely many conjugates of a finitely generated $\Gamma$9 have finite total intersection, the coset condition forces $d(ax_g,x_{ag})<\delta$0 itself to be finite, hence bounded stretch, hence virtual freeness. Consequently, for infinite-index finite-height $d(ax_g,x_{ag})<\delta$1—in particular for quasiconvex subgroups of word-hyperbolic groups, by the Gitik–Mitra–Rips–Sageev width theorem—the coset full shift has ordinary POTP always, and has Cayley-tree POTP if and only if $d(ax_g,x_{ag})<\delta$2 is virtually free. The paper also shows the hypothesis cannot be dropped: for $d(ax_g,x_{ag})<\delta$3, the orbital graph is a tree (after removing loops) yet the coset shift has no Cayley-tree POTP, so orbital quasi-isometry type alone does not determine the property without commensuration.

Commensurated regime. For finitely generated commensurated $d(ax_g,x_{ag})<\delta$4, the main classification theorem gives four equivalent conditions: (i) $d(ax_g,x_{ag})<\delta$5 has Cayley-tree POTP; (ii) the orbital coset graph $d(ax_g,x_{ag})<\delta$6 is a quasi-tree; (iii) the reduced Schlichting completion $d(ax_g,x_{ag})<\delta$7 acts continuously, properly, and cocompactly on a locally finite tree; (iv) $d(ax_g,x_{ag})<\delta$8 is the fundamental group of a finite graph of groups with all vertex and edge groups commensurable with $d(ax_g,x_{ag})<\delta$9. The proof of (i)⇒(ii) builds a tree decomposition of the orbital graph with uniformly bounded outer diameter from convex hulls of coset fibres, then applies Berger–Seymour; the converse lifts an abstract uniform tree on the coset space to a Cayley spanning tree whose replacement paths lie in finitely many $T\subseteq C_S$0-cosets. The implication (ii)⇔(iii) uses the locally compact Milnor–Švarc lemma and a case analysis (compact, two-ended, non-elementary bushy) for Cayley–Abels graphs. The result yields a finite relative tree decomposition rather than a single splitting; an example ($T\subseteq C_S$1 inside $T\subseteq C_S$2) shows that even a nontrivial splitting over $T\subseteq C_S$3 does not suffice.

Virtual cohomological codimension one

Combining the classification with Margolis's codimension-one theorem and Brown's cellular cohomological-dimension inequality, the paper proves: for $T\subseteq C_S$4 and $T\subseteq C_S$5 of type VFP over $T\subseteq C_S$6 with $T\subseteq C_S$7 commensurated of infinite index, the coset full shift has ordinary POTP, and has Cayley-tree POTP if and only if $T\subseteq C_S$8. The forward direction uses Margolis's graph-of-groups decomposition; the reverse bounds $T\subseteq C_S$9 by $(H,\delta)$0 via the tree action, and rules out equality $(H,\delta)$1 because Margolis's finite-index criterion would contradict infinite index. This is a sharp dichotomy: ordinary POTP holds for every such shift, while Cayley-tree POTP isolates exactly the codimension-one case, including nonnormal commensurated subgroups.

Finite-index STRP permanence and genericity

The paper proves that the strong topological Rokhlin property (STRP)—existence of an action with comeager conjugacy class in the Polish space of Cantor actions—passes from a finite-index subgroup $(H,\delta)$2 to a finitely generated overgroup $(H,\delta)$3. The proof constructs, from an $(H,\delta)$4-subshift, a $(H,\delta)$5-subshift with an auxiliary coset-label coordinate, shows this construction preserves projective isolation via an equivariant block map, and uses density of projectively isolated subshifts (Doucha's characterization of STRP). Consequences: every finitely generated virtually free group has STRP, answering the virtually cyclic case of Doucha's commensurability question; and since Doucha showed ordinary POTP is generic for STRP groups while Theorem (i)–(iv) makes ordinary and Cayley-tree POTP coincide for virtually free groups, Cayley-tree POTP is comeager among Cantor actions of every finitely generated virtually free group.

Limitations and open questions

The paper is explicit about the boundaries of its results. The finite-index-overgroup direction of Doucha's STRP permanence question is proved, but the finite-index-subgroup direction—and hence full commensurability invariance of STRP—remains open. Without commensuration, the quasi-isometry type of a single orbital graph does not determine Cayley-tree POTP, and the paper does not offer a replacement criterion in that generality. The finite-height and commensurated regimes are treated separately, and a unified analysis of the coset-finiteness condition for general period subgroups, depending on how $(H,\delta)$6 meets its conjugates, is not provided. The characterization of Cayley-tree POTP for zero-dimensional actions requires zero-dimensionality; behavior for general compact metric spaces is not addressed beyond the necessary equicontinuity condition.

Conclusion

The paper establishes Cayley-tree POTP as a dynamical invariant that detects virtual freeness for full shifts, relative quasi-tree geometry for commensurated pairs, and virtual cohomological codimension one for VFP pairs, while ordinary POTP holds throughout these classes. The replacement-path equicontinuity criterion, expressed through common left-period subgroups, is the unifying mechanism, and the finite-index permanence of STRP yields both the genericity of Cayley-tree POTP for virtually free Cantor actions and the resolution of the virtually cyclic case of Doucha's question.