On Ulam widths of finitely presented infinite simple groups
James Hyde, Yash Lodha
TL;DR
The paper addresses the problem of Ulam width for finitely presented infinite simple groups by constructing, for each $n>1$, a finitely presented infinite simple group $\Gamma_n'$ that is uniformly simple and of type $F_infty$, with width $R(\Gamma_n')$ at least $\lfloor n/2\rfloor$ and unbounded across the family. The authors develop a novel construction using groups $\Omega_n$ and circle actions, rooted in Higman–Thompson groups $F_{2^n}$, to prove uniform simplicity via a detailed analysis of derivative cocycles and orbit structures on $\mathbf{Z}[1/2]$ and the circle. They further establish that the commutator width $P(\Gamma_n')$ is unbounded in the family and show that each group embeds into Thompson’s group $T$, implying absence of property (T) and presence of Haagerup property. Additionally, the groups are shown to be of type $F_infty$, ensuring strong finiteness properties and yielding pairwise non-elementarily equivalent examples among finitely presented infinite simple groups. Overall, the work provides the first explicit finitely presented infinite simple groups with unbounded Ulam width and analyzes their uniform perfection properties, contributing new techniques for proving uniform simplicity in dynamical group actions.
Abstract
A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from $1947$ is uniform simplicity. A group $G$ is said to be $n$-uniformly simple for $n \in \mathbf{N}$ if for every $f,g\in G\setminus \{id\}$, there is a product of no more than $n$ conjugates of $g$ and $g^{-1}$ that equals $f$. Then $G$ is uniformly simple if it is $n$-uniformly simple for some $n \in \mathbf{N}$, and we refer to the smallest such $n$ as the Ulam width, denoted as $\mathcal{R}(G)$. If $G$ is simple but not uniformly simple, one declares $\mathcal{R}(G)=\infty$. In this article, we construct for each $n\in \mathbf{N}$, a finitely presented infinite simple group $G$ such that $n<\mathcal{R}(G)<\infty$. These are the first such examples among the class of finitely presented infinite simple groups. For the class of finitely generated (but not finitely presentable) infinite simple groups, the existence of such examples was settled in the work of Muranov. However, this had remained open for the class of finitely presented infinite simple groups. Our examples are also of type $F_{\infty}$, which means that they are fundamental groups of aspherical CW complexes with finitely many cells in each dimension. Uniformly simple groups are in particular uniformly perfect: there is an $n\in \mathbf{N}$ such that every element of the group can be expressed as a product of at most $n$ commutators of elements in the group. We also show that the analogous notion of width for uniform perfection is unbounded for our family of finitely presented infinite simple groups. To our knowledge, this is also the first such family.