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Koichi Oyakawa

Publications and source records attributed to Koichi Oyakawa.

11 recordsLinked to original sources
arXiv2026

Borel complexity for product of trees and commuting partial maps

We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.

math.GR↗
arXiv2026

Infinite graph product of groups II: Analytic properties

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In particular, we provide a new class of convergence groups and identify the if and only if condition for this convergence action to be geometrically finite. We also provide a new class of properly proximal groups, relatively bi-exact groups, and groups with strongly solid group von Neumann algebras.

math.GR↗
arXiv2026

Hyperfiniteness of the boundary action of virtually special groups

We prove that for any countable group acting virtually specially on a CAT(0) cube complex, the orbit equivalence relation induced by its action on the Roller boundary is hyperfinite. This can be considered as a generalization of hyperfiniteness of the boundary action of cubulated hyperbolic groups by Huang-Sabok-Shinko.

math.GR↗
arXiv2026

Graphical small cancellation and hyperfiniteness of boundary actions

We study actions of (infinitely presented) graphical small cancellation groups on the Gromov boundaries of their coned-off Cayley graphs. We show that a class of graphical small cancellation groups, including (infinitely presented) classical small cancellation groups, admit hyperfinite boundary actions, more precisely, the orbit equivalence relation that they induce on the boundaries of the coned-off Cayley graphs is hyperfinite.

math.GR↗
arXiv2026

Infinite graph product of groups I: Geometry of the extension graph

We introduce the extension graph of graph product of groups and study its geometry. This enables us to study properties of graph product by exploiting large scale geometry of its defining graph. In particular, we show that the extension graph is isomorphic to the crossing graph of a canonical quasi-median graph and exhibits the same phenomenon about asymptotic dimension as quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. As an application of the extension graph, we prove relative hyperbolicity of graph-wreath product. This provides a new construction of relatively hyperbolic groups.

math.GR↗
arXiv2025

Strong solidity classification of Coxeter groups

We prove the dichotomy that every Coxeter group either has a strongly solid group von Neumann algebra or contains the product of an infinite cyclic group and a free group of rank 2. This generalizes the same dichotomy for right-angled Coxeter groups by Borst-Caspers. However, our proof is conceptually different, which leads to a significantly streamlined argument. We also provide additional equivalent geometric and group-theoretic characterizations of strong solidity for Coxeter groups that allow us to completely classify those with a strongly solid group von Neumann algebra. In particular, we characterize strong solidity purely in terms of the defining Coxeter-Dynkin diagram. Finally, we obtain the same dichotomy for virtually cocompact special groups.

math.OA↗
arXiv2025

Hyperfiniteness for group actions on trees

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfinitenss of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.

math.GR↗
arXiv2025

Small cancellation groups are bi-exact

We prove that finitely generated (not necessarily finitely presented) $C'(\frac{1}{33})$-groups are bi-exact. This is a new class of bi-exact groups.

math.GR↗
arXiv2024

Bi-exactness of relatively hyperbolic groups

We prove that finitely generated relatively hyperbolic groups are bi-exact if and only if all peripheral subgroups are bi-exact. This is a generalization of Ozawa's result which claims that finitely generated relatively hyperbolic groups are bi-exact if all peripheral subgroups are amenable.

math.GR↗
arXiv2024

Hyperfiniteness of boundary actions of acylindrically hyperbolic groups

We prove that for any countable acylidrically hyperbolic group $G$, there exists a generating set $S$ of $G$ such that the corresponding Cayley graph $Γ(G,S)$ is hyperbolic, $|\partialΓ(G,X)|>2$, the natural action of $G$ on $Γ(G,S)$ is acylindrical, and the natural action of $G$ on the Gromov boundary $\partialΓ(G,S)$ is hyperfinite. This result broadens a class of groups that admit a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action.

math.GR↗