相关论文: Nonamenable subforests of multi-ended quasi-pmp gr…

We study locally countable acyclic measure-class-preserving (mcp) Borel graphs by analyzing their "topography" -- the interaction between the geometry and the associated Radon--Nikodym cocycle. We identify three notions of topographic…

We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a…

We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its…

We prove several results showing that every locally finite Borel graph whose large-scale geometry is "tree-like" induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has…

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

A framework to handle tree decompositions of the components of a Borel graph in a Borel fashion is introduced, along the lines of Tserunyan's Stallings Theorem for equivalence relations arXiv:1805.09506. This setting leads to a notion of…

Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have…

Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct…

We give a cohomological criterion for certain decomposition of Borel graphs, which is an analog of Dunwoody's work on accessibility of groups. As an application, we prove that a Borel graph $(X,G)$ with uniformly bounded degrees of…

We interpret the probabilistic notion of unimodularity for measures on the space of rooted locally finite connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in…

The arboricity $\Gamma$ of a graph is the minimum number of forests its edge set can be partitioned into. Previous approximation schemes were nonconstructive, i.e., they only approximated the arboricity as a value without computing a…

We prove that the infinite components of the Free Uniform Spanning Forest of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the…

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance…

We introduce a graph partitioning problem motivated by computational topology and propose two algorithms that produce approximate solutions. Specifically, given a weighted, undirected graph $G$ and a positive integer $k$, we desire to find…

In the aftermath of the Robertson--Seymour Graph Minor Theorem, Thomas conjectured that the countable graphs are well-quasi-ordered under the minor relation. We prove that this conjecture, when restricted to graphs with no infinite paths…

Graph isomorphism, subgraph isomorphism, and maximum common subgraphs are classical well-investigated objects. Their (parameterized) complexity and efficiently tractable cases have been studied. In the present paper, for a given set of…

We prove several criteria for quasi-isometry between non-locally-finite graphs and their structure trees. Results of M\"oller in \cite{moeller92ends2} for locally finite and transitive graphs are generalized. We also give a criterion which…

This paper studies the structure of graphs with given tree-width and excluding a fixed complete bipartite subgraph, which generalises the bounded degree setting. We give a new structural description of such graphs in terms of so-called…

We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient…

We study a uniform, quantitative form of the amenability-hyperfiniteness paradigm for bounded-degree Borel graphs generating countable Borel equivalence relations. We introduce \emph{uniform Borel amenability} and prove that it is…