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James Belk

Publications and source records attributed to James Belk.

At least 19 recordsLinked to original sources
arXiv2025

Hyperbolic groups satisfy the Boone-Higman conjecture

The 1973 Boone-Higman conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In this paper, we show that hyperbolic groups satisfy this conjecture, that is, each hyperbolic group embeds in some finitely presented simple group. This shows that the conjecture holds in the "generic" case for finitely presented groups. Our key tool is a new family of groups, which we call "rational similarity groups (RSGs)", that is interesting in its own right. We prove that every hyperbolic group embeds in a full, contracting RSG, and every full, contracting RSG embeds in a finitely presented simple group, thus establishing the result. Another consequence of our work is that all contracting self-similar groups satisfy the Boone-Higman conjecture.

math.GR↗
arXiv2025

Compact aspherical solenoids

We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes $P$-adic solenoids, as well as the universal hyperbolic solenoid of Teichmüller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove a Dehn-Nielsen-type theorem for self-homotopy equivalences of such a solenoid. This generalizes a result of Odden regarding the universal hyperbolic solenoid.

math.GT↗
arXiv2025

Progress around the Boone-Higman Conjecture

A conjecture of Boone and Higman from the 1970's asserts that a finitely generated group $G$ has solvable word problem if and only if $G$ can be embedded into a finitely presented simple group. We comment on the history of this conjecture and survey recent results that establish the conjecture for many large classes of interesting groups.

math.GR↗
arXiv2025

The Maximality of $T$ in Thompson's group $V$

We show that R. Thompson's group $T$ is a maximal subgroup of the group $V$. The argument provides examples of foundational calculations which arise when expressing elements of $V$ as products of transpositions of basic clopen sets in Cantor space $\mathfrak{C}$.

math.GR↗
arXiv2025

Boone-Higman embeddings of $\mathrm{Aut}(F_n)$ and mapping class groups of punctured surfaces

We prove that the groups $\mathrm{Aut}(F_n)$ satisfy the Boone-Higman conjecture for all $n$, meaning each $\mathrm{Aut}(F_n)$ embeds in a finitely presented simple group. In fact, we prove that each $\mathrm{Aut}(F_n)$ satisfies the "permutational" Boone-Higman conjecture, which means the simple group in question can be taken to be a twisted Brin-Thompson group. A far-reaching consequence of our approach is that finitely presented twisted Brin-Thompson groups are universal among finitely presented simple groups that are highly transitive. This is evidence toward the Boone-Higman conjecture being equivalent to its permutational version. Proving the conjecture for $\mathrm{Aut}(F_n)$ also confirms the conjecture for all groups (virtually) embedding into some $\mathrm{Aut}(F_n)$, such as mapping class groups of non-closed surfaces, braid groups, loop braid groups, ribbon braid groups and certain Artin groups. This answers several questions of the first and fourth authors with Bleak and Matucci. Yet another consequence of our approach is that satisfying the permutational Boone-Higman conjecture is closed under free products.

math.GR↗
arXiv2024

Finite Germ Extensions

We prove finiteness properties for groups of homeomorphisms that have finitely many "singular points", and we describe the normal structure of such groups. As an application, we prove that every countable abelian group can be embedded into a finitely presented simple group, verifying the Boone-Higman conjecture for countable abelian groups. Indeed, we describe a specific 2-generated, $\mathrm{F}_\infty$ simple group $V\mathcal{A}$ of homeomorphisms of the Cantor set that contains every countable abelian group. As a second application, we prove that if $G$ is a bounded automata group then the associated Röver-Nekrashevych groups $V_{d,r}G$ have type $\mathrm{F}_\infty$, verifying a conjecture of Nekrashevych for a large class of contracting self-similar groups. Among others, this result applies to Röver-Nekrashevych groups associated to Gupta-Sidki groups and the basilica group.

math.GR↗
arXiv2024

Boone--Higman Embeddings for Contracting Self-Similar Groups

We give a short proof that every contracting self-similar group embeds into a finitely presented simple group. In particular, any contracting self-similar group embeds into the corresponding Röver--Nekrashevych group, and this in turn embeds into one of the twisted Brin--Thompson groups introduced by the first author and Matthew Zaremsky. The proof here is a simplification of a more general argument given by the authors, Collin Bleak, and Matthew Zaremsky for contracting rational similarity groups.

math.GR↗
arXiv2024

Type systems and maximal subgroups of Thompson's group $V$

We introduce the concept of a type system~$\Part$, that is, a partition on the set of finite words over the alphabet~$\{0,1\}$ compatible with the partial action of Thompson's group~$V$, and associate a subgroup~$\Stab{V}{\Part}$ of~$V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of~$V$. We also find an uncountable family of pairwise non-isomorphic maximal subgroups of~$V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$ of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$ (both related to primitivity) are each satisfied only by $V$ itself, giving new ways to recognise when a subgroup of $V$ is not actually proper.

math.GR↗
arXiv2024

Quasisymmetries of finitely ramified Julia sets

We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of "undistorted metrics" that are all quasi-equivalent. As a result, piecewise-defined homeomorphisms of such a fractal that locally preserve the cell structure are quasisymmetries. This immediately gives a solution to the quasisymmetric uniformization problem for topologically rigid fractals such as the Sierpiński triangle. We show that our theory applies to many finitely ramified Julia sets, and we prove that any connected Julia set for a hyperbolic unicritical polynomial has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove that the quasisymmetry group of the Julia set for the rational function $1-z^{-2}$ is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson's group $F$.

math.DS↗
arXiv2023

Rearrangement groups of fractals

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson's groups F, T, and V . Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite CAT(0) cubical complex, and we use this action to prove that certain rearrangement groups are of type F infinity.

math.GR↗
arXiv2023

A Thompson group for the basilica

We describe a Thompson-like group of homeomorphisms of the basilica Julia set. Each element of this group acts as a piecewise-linear homeomorphism of the unit circle that preserves the invariant lamination for the basilica. We develop an analogue of tree pair diagrams for this group which we call arc pair diagrams, and we use these diagrams to prove that the group is finitely generated. We also prove that the group is virtually simple.

math.GR↗
arXiv2023

Embedding hyperbolic groups into finitely presented infinite simple groups

The Boone--Higman conjecture is that every recursively presented group with solvable word problem embeds in a finitely presented simple group. We discuss a brief history of this conjecture and work towards it. Along the way we describe some classes of finitely presented simple groups, and we briefly outline work of Belk, Bleak, Matucci, and Zaremsky showing that the broad class of hyperbolic groups embeds in a class of finitely presented simple groups.

math.GR↗
arXiv2023

Pseudo-$F_4$ is isomorphic to $F_4$

We prove that the "pseudo-$F_4$" group is isomorphic to $F_4$, answering a question of Brin. Both of these groups can be described as fast groups of homeomorphisms of the interval generated by bumps, as introduced by Bleak, Brin, Kassabov, Moore, and Zaremsky. The proof uses a representation of fast groups as Guba-Sapir diagram groups in order to leverage known results on isomorphisms of diagram groups.

math.GR↗
arXiv2022

Conjugator length in Thompson's groups

We prove Thompson's group $F$ has quadratic conjugator length function. That is, for any two conjugate elements of $F$ of length $n$ or less, there exists an element of $F$ of length $O(n^2)$ that conjugates one to the other. Moreover, there exist conjugate pairs of elements of $F$ of length at most $n$ such that the shortest conjugator between them has length $Ω(n^2)$. This latter statement holds for $T$ and $V$ as well.

math.GR↗
arXiv2022

A piecewise linear homeomorphism of the circle which is periodic under renormalization

We demonstrate the existence of a piecewise linear homeomorphism $f$ of $\mathbb{R}/\mathbb{Z}$ which maps rationals to rationals, whose slopes are powers of $\frac{2}{3}$, and whose rotation number is $\sqrt{2}-1$. This is achieved by showing that a renormalization procedure becomes periodic when applied to $f$. Our construction gives a negative answer to a question of D. Calegari. When combined with work of the 2nd and 3rd authors, our result also shows that $F_{\frac{2}{3}}$ does not embed into $F$, where $F_{\frac{2}{3}}$ is the subgroup of the Stein-Thompson group $F_{2,3}$ consisting of those elements whose slopes are powers of $\frac{2}{3}$. Finally, we produce some evidence suggesting a positive answer to a variation of Calegari's question and record a number of computational observations.

math.GR↗
arXiv2022

Embedding $\mathbb{Q}$ into a Finitely Presented Group

We observe that the group of all lifts of elements of Thompson's group $T$ to the real line is finitely presented and contains the additive group $\mathbb{Q}$ of the rational numbers. This gives an explicit realization of the Higman embedding theorem for $\mathbb{Q}$, answering a Kourovka notebook question of Martin Bridson and Pierre de la Harpe.

math.GR↗
arXiv2022

A short proof of Rubin's theorem

In a remarkable theorem, M. Rubin proved that if a group $G$ acts in a locally dense way on a locally compact Hausdorff space $X$ without isolated points, then the space $X$ and the action of $G$ on $X$ are unique up to $G$-equivariant homeomorphism. Here we give a short, self-contained proof of Rubin's theorem, using equivalence classes of ultrafilters on a poset to reconstruct the points of the space $X$.

math.GR↗