arXiv · 2610.18360
The goal of this note is to provide a human written exposition of OpenAI's recent proof that Thompson's group F is nonamenable.
Explore connections, maps & timelines
Sang-hyun Kim, Yash Lodha. 2026-10-13. A note on the proof of nonamenability of F. https://arxiv.org/abs/2610.18360
Cite the original work for its findings. Save a collection to share your selection of sources.
KEEP EXPLORING
We develop a fine-scale structural theory for balls of polynomial volume in finitely generated groups. Our main result shows that a polynomial-volume hypothesis on a single ball implies that every larger ball can, after quotienting by a certain canonical finite normal subgroup, be embedded in a lattice of a Lie group whose connected component is nilpotent with sharply bounded index and dimension. These Lie groups and lattices form chains of successive quotients that are essentially unique, giving a quantitative fine-scale analogue of Losert's strengthening of the Mal'cev completion. This describes the fine structure of the group with sharp algebraic and quantitative precision; for example, it leads to sharp bounds in Breuillard, Green and Tao's finitary version of Gromov's polynomial-growth theorem. These results have a wide range of applications in various aspects of the theory of vertex-transitive graphs, including for instance random walks, percolation, and scaling limits. We obtain some of these applications in the present paper, and treat others in companion papers. Some are due to or joint with other authors.
Rota-Baxter Lie groups are the group-level counterparts of Rota-Baxter Lie algebras of weight $1$. In this paper, we first establish a correspondence between representations of Rota-Baxter Lie groups and representations of Rota-Baxter Lie algebras of weight $1$ via differentiation and integration. We then develop a cohomology theory for Rota-Baxter Lie groups with coefficients in a representation by constructing an explicit cochain complex in arbitrary degrees. We justify this theory by proving a van Est type theorem, which relates the cohomology of a Rota-Baxter Lie group to that of its Rota-Baxter Lie algebra of weight $1$. As an application, we prove that equivalence classes of central extensions of a Rota-Baxter Lie group are classified by the second cohomology group.
A lazy cellular automaton $τ:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity $e$, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that, over every group, the existence of a compatible set is necessary for $τ$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \in S_a$, $s \in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.