The Complexity of Proper Homotopy Equivalence of Graphs

TL;DR

The paper investigates proper homotopy equivalence (PHE) of locally finite graphs, proving that PHE is Borel complete by a reduction to the universal orbit equivalence $C_\infty$, and showing the Loch Ness monster PHE class is comeager among infinite graphs. It develops standard Borel spaces for graphs, with end-space data $(E(\Gamma),E_{\ell}(\Gamma))$ that classify PHE together with rank, and leverages Camerlo–Gao to connect to $C_\infty$ completeness. The authors extend the results to bounded-degree and locally finite graph spaces, using marked-graph spaces and inverse-limit representations to establish Borel reducibility between PHE in these spaces. They transfer the entire framework to noncompact surfaces with pants decompositions, proving the homeomorphism relation on these surfaces is likewise $C_\infty$-complete and that Loch Ness monster-type endclasses are comeager, with the end-space map being Borel. These findings provide a sharp descriptive-set-theoretic description of infinite-type symmetry groups in both graphs and surfaces, highlighting the central role of end-space data in classification and complexity.

Abstract

We demonstrate that the proper homotopy equivalence relation for locally finite graphs is Borel complete. Furthermore, among the infinite graphs, there is a comeager equivalence class. As corollaries, we obtain the analogous results for the homeomorphism relation of noncompact surfaces with pants decompositions.

Figures (10)

• Figure 1: Representations of the Loch Ness monster graph (top) and surface (bottom).
• Figure 2: An example of a graph where $v_1$ has two self loops, and there are multiple edges between $v_2$ and $v_3$. The degree of $v_1$ is $5$, the degrees of $v_2$ and $v_3$ are $4$, and the degree of $v_4$ is $1$.
• Figure 3: Examples of graphs with different PHE types.
• Figure 4: Two graphs which are proper homotopy equivalent. The left graph is $6$-regular, while right graph is $3$-regular.
• Figure 5: For the black graph ${\Gamma}$, the ball of radius $2$ is drawn in blue. To find a one-ended graph in $\mathcal{U}_{({\Gamma},2)}$ we add the orange edge.

Theorems & Definitions (47)

• Example 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Theorem 2.2: ayala1990proper
• Proposition 2.3
• proof
• proof
• Proposition 2.5