Measurable One-Ended Spanning Subtree

• Measurable one-ended spanning subtree is a definable acyclic spanning subgraph of an infinite graph where each component has exactly one end.
• The construction relies on various definability regimes such as invariant, factor-of-iid, and Borel settings to ensure measurable outcomes.
• Hyperfiniteness is identified as the precise measure-theoretic criterion for the existence of one-ended spanning subtrees almost everywhere.

Measurable one-ended spanning subtree is a broad label for results asserting that an infinite graph admits an acyclic spanning subgraph whose relevant components are trees with exactly one end, subject to a definability requirement such as invariance, factor-of-i.i.d., Borelness on a comeager or conull set, or measurability almost everywhere. In the cited literature the precise objects are usually called a one-ended spanning tree when each connected component is spanned by a tree, or a one-ended spanning subforest when the measurable output need not remain connected on each original component. Classically, Halin proved that an infinite locally finite graph admits a one-ended spanning tree iff the graph itself is one-ended; measurable versions retain the same geometric target but introduce additional structural obstructions and several distinct ambient categories (Bowen et al., 18 Sep 2025, Bowen et al., 2022, Timar, 2018).

1. Basic notion and graph-theoretic content

A connected graph is one-ended if after removing any finite set of vertices, there remains exactly one infinite connected component. For trees, equivalent formulations used in the literature include: there is exactly one equivalence class of infinite rays; from every vertex, there is a unique ray to infinity up to finite overlap; and every vertex separates off only finite pieces from infinity. A nonconnected graph is one-ended if each connected component is one-ended (Timar, 2018, Bowen et al., 2022).

A spanning tree of a graph $G$ means a subgraph $T\subseteq G$ such that in each connected component of $G$, the induced subgraph $T$ on that component is a tree spanning all vertices of that component. A one-ended spanning tree therefore requires each such tree component to have one end. Since any spanning tree is in particular a spanning subtree, tree-existence theorems automatically imply spanning-subtree theorems. In measurable settings the term spanning forest is often used because the output is allowed to be componentwise, or because the measurable construction does not always preserve one tree per original component (Bowen et al., 2022, Conley et al., 2021).

The central measurable difficulty is that one-endedness of the ambient graph is not by itself enough. In the measurable world, the existence problem depends on the mode of definability and, in several settings, on additional hypotheses such as amenability, hyperfiniteness, or the absence of two-ended components. This suggests that the measurable theory is not a straightforward transcription of the classical graph-theoretic statement (Bowen et al., 18 Sep 2025, Conley et al., 2021).

2. Definability regimes

The subject splits according to the ambient category and the meaning of “measurable.” In the unimodular setting, the relevant notions are invariant, factor of iid, and jointly unimodular. In the Borel-graph setting on Polish spaces, the main regimes are generically on an invariant comeager Borel set, $\mu$-a.e. on an invariant $\mu$-conull Borel set, and everywhere on all of the space. In measurable graphings on standard probability spaces, the key distinction is between pmp and mcp graphs, with conclusions stated almost everywhere (Timar, 2018, Bowen et al., 2022, Bowen et al., 18 Sep 2025).

Regime	Typical output
Unimodular / Cayley	invariant or factor-of-iid one-ended spanning tree
Borel graph on Polish space	Borel one-ended spanning tree generically, a.e., or everywhere
Measurable graphing on probability space	measurable one-ended spanning tree or Borel a.e. one-ended spanning subforest

A percolation in the unimodular language means a random subgraph whose distribution is jointly unimodular with the ambient graph. A factor of iid means the subgraph is obtained as an equivariant Borel function of i.i.d. $[0,1]$-labels on vertices. In the Borel setting, the literature is more precise than the generic phrase “measurable subtree”: the output is usually an actual Borel subgraph after restricting to a suitable invariant domain, rather than merely an equivalence class modulo null or meager sets (Timar, 2018, Bowen et al., 2022).

3. Amenable unimodular random graphs and invariant trees

A decisive result for the invariant and unimodular theory is the theorem that if $G$ is an ergodic amenable unimodular random graph that has one end almost surely, then there is a factor-of-iid spanning tree of $G$ that has one end almost surely. For quasi-transitive unimodular graphs, amenability and one-endedness are equivalent to the existence of an invariant spanning tree with one end; in particular, every amenable one-ended Cayley graph has an invariant one-ended spanning tree (Timar, 2018).

This sharpens earlier constructions of Benjamini–Lyons–Peres–Schramm and Aldous–Lyons, which produced measurable spanning trees with 1 or 2 ends but did not control which case occurred. The novelty is exactly the measurable forcing of one end when the ambient graph is itself amenable and one-ended. The result is strong both combinatorially and measurably: the output is spanning, connected, acyclic, hence a spanning tree, and it has exactly one end almost surely (Timar, 2018).

The proof has two stages. The reduction lemma states that if $G$ admits a factor-of-i.i.d. sequence $T\subseteq G$0 of connected subgraphs with

$T\subseteq G$1

then $T\subseteq G$2 has a one-ended factor-of-i.i.d. spanning tree. This converts “rare connected cores” into a one-ended spanning tree by constructing increasing forests $T\subseteq G$3 with finite components and taking a limit $T\subseteq G$4. Connectivity is forced by estimates of the form

$T\subseteq G$5

while one-endedness is verified by showing that each vertex is separated from infinity by one point in the final tree (Timar, 2018).

The second stage produces the shrinking connected subgraphs $T\subseteq G$6. Starting from a known factor-of-i.i.d. spanning tree with 1 or 2 ends, the only difficult case is a 2-ended tree with a unique bi-infinite trunk $T\subseteq G$7. The proof projects the ambient graph onto a graph $T\subseteq G$8 on the trunk and establishes the sublinear-distance lemma

$T\subseteq G$9

This allows sparse Bernoulli points on the trunk to be connected by finite connectors whose density tends to zero. Lifting these connectors back to the original graph yields the required $G$0, and the reduction lemma then produces the final one-ended spanning tree (Timar, 2018).

4. Borel one-ended spanning trees and connected toast

In the Borel setting, a major synthesis is the theorem that any locally finite one-ended Borel graph on a Polish space admits a Borel one-ended spanning tree on an invariant comeager set; if the graph is induced by a free Borel action of an amenable group, the same conclusion holds on an invariant $G$1-conull set; and if it is induced by a free Borel action of a polynomial growth group, the tree exists everywhere (Bowen et al., 2022).

The organizing concept is connected toast. A Borel collection $G$2 is a toast when every edge of the graph lies inside some tile and tiles satisfy a controlled nesting/disjointness condition; it is connected when the induced subgraph on the “new part”

$G$3

is connected for every tile $G$4. Proposition 2.2 in that work states that if $G$5 admits a connected toast, then it admits a Borel one-ended spanning tree. The tree is assembled explicitly by taking a finite tree on each fresh layer and then adding one outward edge from each tile, so that the nesting of tiles witnesses a unique direction to infinity (Bowen et al., 2022).

The three toast theorems correspond to the three definability regimes. Generic existence for arbitrary locally finite one-ended Borel graphs is obtained by a Baire category argument refining a pre-existing Borel toast. The conull theorem for free actions of amenable groups is based on connected Følner-type tiles and a measurable exhaustion. The everywhere theorem for free actions of polynomial growth groups uses finite Borel asymptotic dimension and a uniform boundary-connectivity statement for one-ended finitely presented groups (Bowen et al., 2022).

These spanning-tree results feed into definable combinatorics. The same connected-toast machinery yields Borel balanced orientations in even-degree graphs and perfect matching theorems, including the statement that bipartite one-ended $G$6-regular Borel graphs admit Baire measurable perfect matchings. A plausible implication is that one-ended spanning trees are functioning here as a canonical definable skeleton rather than merely as isolated existence statements (Bowen et al., 2022).

5. One-ended spanning subforests, planar duality, and treeability

A different branch of the subject develops one-ended spanning subforests as the primary measurable object. For locally finite acyclic Borel graphs with no connected components of $G$7 or $G$8 ends, one obtains, for every Borel probability measure $G$9, a $T$0-conull Borel set $T$1 and a one-ended Borel function $T$2 whose graph is contained in $T$3; likewise, for every compatible Polish topology, one obtains such a function on a comeager Borel set. Equivalently, on a conull or comeager invariant domain the graph contains a Borel one-ended spanning subforest (Conley et al., 2016).

The technical innovation there is the notion of an ample acyclic graph: every vertex has degree at least $T$4, and after removing any vertex, every remaining component contains a vertex of degree at least $T$5. Marker-set orientations, decreasing sequences of marker sets, and an iterative sparse-removal argument in ample graphs together produce one-ended Borel functions. These one-ended subforests are then used as a recursive framework for measurable list coloring and, ultimately, for measurable Brooks-type coloring results (Conley et al., 2016).

For p.m.p. graphings the theory becomes classification-theoretic. An aperiodic locally finite p.m.p. Borel graph $T$6 has a Borel almost-everywhere one-ended spanning subforest iff $T$7 is $T$8-nowhere two-ended. The same work proves hyperfinite and acyclic special cases, shows that in the hyperfinite p.m.p. regime one can often preserve the connectedness relation almost everywhere, and identifies one-ended spanning subforests as the key input for measurable treeings (Conley et al., 2021).

In planar Borel graphs, the central mechanism is duality. Given a Borel $T$9-basis $\mu$0 and dual graph $\mu$1, a one-ended spanning subforest $\mu$2 determines an acyclic primal subgraph

$\mu$3

The duality theorem states that this primal graph is acyclic and has the same connected components as $\mu$4 iff $\mu$5 is a one-ended spanning subforest of the dual. That construction is then used to establish measure treeability for Borel planar graphs and measure strong treeability for groups admitting planar geometric models, including finitely generated groups with planar Cayley graphs, $\mu$6 and its closed subgroups, and finitely generated elementarily free groups (Conley et al., 2021).

6. Hyperfiniteness as the exact measurable criterion

The most complete characterization currently described in the supplied literature is the theorem that a locally finite, one-ended, measure-class-preserving measurable graph on a standard probability space admits a measurable one-ended spanning tree almost everywhere iff it is hyperfinite. Equivalently,

$\mu$7

under the assumptions that $\mu$8 is locally finite, one-ended, and mcp (Bowen et al., 18 Sep 2025).

This identifies measure-hyperfiniteness as the exact measurable obstruction. Necessity is conceptually straightforward: a one-ended locally finite tree can be exhausted by iteratively removing leaves, and therefore is hyperfinite. Sufficiency is much more elaborate. The argument introduces substantial subgraphs, meaning measurable subgraphs whose components are either one-ended or isolated and which preserve exactly one one-ended core inside each one-ended component of the ambient graph. A basic proposition reduces the spanning-tree problem to finding a nested sequence of substantial measurable subgraphs

$\mu$9

From such a shrinking sequence, the final measurable one-ended spanning tree is reconstructed by measurably attaching isolated vertices back to the one-ended core by finite trees (Bowen et al., 18 Sep 2025).

The shrinking argument proceeds through successive structural reductions. First, almost all edges outside a measurable spanning tree can be deleted. Next, one studies the core of the spanning tree, defined by those tree edges lying in infinitely many fundamental cycles relative to the ambient graph. Non-core edges can largely be removed while preserving substantiality, and after contracting finite non-core pieces one may assume that the entire spanning tree is core. In that regime the graph is sufficiently edge-connected to permit finite-graph arguments: after further reductions every nontrivial component is $\mu$0-edge-connected, and the Nash–Williams theorem on edge-disjoint spanning trees yields a substantial subgraph with uniformly smaller edge measure. Iterating produces the required $\mu$1 sequence (Bowen et al., 18 Sep 2025).

This theorem extends prior pmp and amenable-action results to the full measure-class-preserving setting and answers a question of Bowen–Poulin–Zomback. It also gives a graphing-level explanation of earlier invariant results: via cluster graphings, one can derive that if $\mu$2 is connected, locally finite, one-ended, and quasi-transitive, then amenability of the acting group is equivalent to the existence of a $\mu$3-invariant random spanning tree of $\mu$4 with one end almost surely, and moreover it can be made factor of i.i.d. (Bowen et al., 18 Sep 2025).

7. Scope, limitations, and structural contrasts

The existence theory is sharp within each regime but not uniform across all settings. The unimodular theorem is restricted to ergodic amenable unimodular random graphs that are one-ended almost surely, hence to amenable one-ended Cayley and quasi-transitive unimodular graphs; it does not claim analogous statements for nonamenable unimodular random graphs, graphs with more than one end, arbitrary nonunimodular graphs, or arbitrary Borel graphings beyond the unimodular and invariant setup. In the quasi-transitive amenable case, the one-endedness assumption is necessary: if such a graph has two ends, then all its invariant spanning trees are two-ended (Timar, 2018).

The Borel theory also has clear boundaries. The definable-combinatorics results establish generic existence for all locally finite one-ended Borel graphs, conull existence for free amenable actions, and everywhere existence for free polynomial-growth actions, but they do not prove an everywhere theorem for all one-ended hyperfinite Borel graphs. One open question stated there is whether every locally finite, one-ended, hyperfinite Borel graph also admits a Borel one-ended spanning tree a.e. In a related direction, the p.m.p. subforest classification suggests a broader nonsingular conjectural picture, but outside the proved hyperfinite, acyclic, and planar cases a full non-p.m.p. characterization was not yet available in that work (Bowen et al., 2022, Conley et al., 2021).

A common misconception is that one-endedness of the ambient graph should force one-endedness for all canonical spanning connected subgraphs. The literature shows the opposite. In one-ended finitely generated abelian Cayley graphs with infinite-order generators, the entire edge set can be decomposed into edge-disjoint Hamiltonian double-rays; each Hamiltonian double-ray is spanning, connected, and exactly two-ended. This provides a sharp contrast object: controlled spanning subgraphs in one-ended ambient graphs need not inherit one-endedness, and measurable one-ended spanning-tree theorems therefore require genuine additional structure rather than only ambient end data (Erde et al., 2017).

Taken together, the supplied results indicate a coherent hierarchy. Invariant and factor-of-i.i.d. one-ended spanning trees are available for amenable one-ended unimodular graphs; Borel one-ended spanning trees exist generically, conullly, or everywhere in several one-ended Borel settings; Borel almost-everywhere one-ended spanning subforests admit a p.m.p. classification by the absence of two-ended components; and for locally finite one-ended measurable graphs on standard probability spaces, hyperfiniteness is exactly the obstruction to measurable one-ended spanning trees (Timar, 2018, Bowen et al., 2022, Conley et al., 2021, Bowen et al., 18 Sep 2025).