arXiv subjects

Henry Bradford

Publications and source records attributed to Henry Bradford.

At least 19 recordsLinked to original sources

arXiv2026

Co-intersection graphs of nonabelian finite simple groups have diameter two

Let $G$ be a finite group. The co-intersection graph $Δ_G ^c$ of $G$ has as its vertices the nontrivial proper subgroups of $G$, with edges joining those pairs of subgroups which intersect trivially. It is clear that every connected component of $Δ_G ^c$ has diameter at most three. In this Note, we show that when $T$ is a nonabelian finite simple group, $Δ_T$ is connected of diameter exactly two. We also make several observations about the extent to which a finite group is determined by its co-intersection graph, and about the class of finite graphs arising as co-intersection graphs.

math.GR↗

arXiv2026

Mixed identities for simple locally finite groups

A mixed identity of a group is a nontrivial word with constants that vanishes under every substitution of its variables. We derive lower bounds for the length of mixed identities in finite simple groups of Lie type, and characterise exactly those families of such groups of bounded rank which satisfy mixed identities of bounded length. We classify the infinite simple locally finite groups admitting a mixed identity: apart from an explicit list of alternating, finitary linear classical, and non-simply-laced groups of Lie type, no such group exists. For the groups in this list, we determine when mixed identities must be singular and obtain restrictions on their critical constants. Moreover, we prove that simple compact Lie groups do not admit mixed identities.

math.GR↗

arXiv2026

Groups with arbitrarily poor permutation stability

We propose a quantitative notion of permutation stability for finitely generated groups. Our notion is related to, but distinct from, the ``stability rate'' introduced by Becker and Mosheiff (which is valid within the class of finitely presented groups). We construct a family of finitely generated stable groups which exhibit, quantitatively, arbitrarily ``bad'' permutation stability. This means that any application of a ``sample-and-substitute'' algorithm will be very slow in ascertaining whether a given tuple of permutations satisfy the defining relations of our groups.

math.GR↗

arXiv2026

Groups of arbitrary lawlessness growth

For a finitely generated lawless group $Γ$ and $n \in \mathbb{N}$, let $\mathcal{A}_Γ (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed rank $k \geq 2$, there exists $\overline{g} \in Γ^k$ of word-length at most $M_n$ with $w(\overline{g}) \neq e$. For any unbounded nondecreasing function $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying some mild assumptions, we construct $Γ$ such that the function $\mathcal{A}_Γ$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_Γ$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_Γ$ grows faster than any tower of exponential functions.

math.GR↗

arXiv2026

Sofic actions, halo products, and metric approximations of groups

We introduce the notion of a ``sofic $\mathcal{C}$-action'' of one group on another by automorphisms, for $\mathcal{C}$ a class of groups. We show that if $\mathcal{C}$ is the class of (i) sofic, (ii) hyperlinear, (iii) linear sofic or (iv) weakly sofic groups, then the class $\mathcal{C}$ is closed under taking semidirect products with sofic $\mathcal{C}$-action. We use this to construct a wide variety of new examples of groups in the classes (i)-(iv), many of them arising as ``halo products'' in the sense of Genevois-Tessera. We have a parallel set of results producing new examples of semidirect products which are locally embeddable into finite groups. Our framework also unifies existing results in the literature, due to Hayes-Sale; Brude-Sasyk and Gao-Kunnawalkam Elayavalli-Patchell.

math.GR↗

arXiv2026

Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks

A mixed equation in a group $G$ is given by a non-trivial element $w (x)$ of the free product $G \ast \mathbb{Z}$, and a solution is some $g\in G$ such that $w(g)$ is the identity. For $G$ acylindrically hyperbolic with trivial finite radical (e.g. torsion-free) we show that any mixed equation of length $n$ has a non-solution of length comparable to $\log(n)$, which is the best possible bound. Similarly, we show that there is a common non-solution of length $O(n)$ to all mixed equations of length $n$, again the best possible bound. In fact, in both cases we show that a random walk of appropriate length yields a non-solution with positive probability.

math.GR↗

arXiv2024

Local permutation stability

We introduce a notion of "local stability in permutations" for finitely generated groups. If a group is sofic and locally stable in our sense, then it is also locally embeddable into finite groups (LEF). Our notion is weaker than the "permutation stability" introduced by Glebsky-Rivera and Arzhantseva-Paunescu, which allows one to upgrade soficity to residual finiteness. We prove a necessary and sufficient condition for an amenable group to be locally permutation stable, in terms of invariant random subgroups (IRSs), inspired by a similar criterion for permutation stability due to Becker, Lubotzky and Thom. We apply our criterion to prove that derived subgroups of topological full groups of Cantor minimal subshifts are locally stable, using Zheng's classification of IRSs for these groups. This last result provides continuum-many groups which are locally stable, but not stable.

math.GR↗

arXiv2024

Hopfian wreath products and the stable finiteness conjecture

We study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. Our main result establishes a strong connection between this problem and Kaplansky's stable finiteness conjecture. Namely, the latter holds true if and only if for every finitely generated abelian group $A$ and every finitely generated Hopfian group $Γ$ the wreath product $A \wr Γ$ is Hopfian. In fact, we characterize precisely when $A \wr Γ$ is Hopfian, in terms of the existence of one-sided units in certain matrix algebras over $\mathbb{F}_p[Γ]$, for every prime $p$ occurring as the order of some element in $A$. A tool in our arguments is the fact that fields of positive characteristic locally embed into matrix algebras over $\mathbb{F}_p$ thus reducing the stable finiteness conjecture to the case of $\mathbb{F}_p$. A further application of this result shows that the validity of Kaplansky's stable finiteness conjecture is equivalent to a version of Gottschalk's surjunctivity conjecture for additive cellular automata.

math.GR↗

arXiv2024

On the spectrum of residual finiteness growth functions

In [K. Bou-Rabee, B. Seward, J. Reine Angwe. Math. 2016] Bou-Rabee and Seward constructed examples of finitely generated residually finite groups $G$ whose residual finiteness growth function $\mathcal{F}_G$ can be at least as fast as any prescribed function. In this note we describe a modified version of their construction, which allows us to give a complementary upper bound on $\mathcal{F}_G$. As such, every nondecreasing function at least $\exp ( n \log (n)^2 \log \log (n)^{1+ε} )$ is close to the residual finiteness growth function of some finitely generated group. We also have similar result for the full residual finiteness growth function and for the divisibility function.

math.GR↗

arXiv2023

Non-singular word maps for linear groups

We study the word image of words with constants in ${\rm GL}(V)$ and show that it is large provided the word satisfies some natural conditions on its length and its critical constants. There are various consequences: We prove that for every $l \geq 1$, there are only finitely many pairs $(n,q)$ such that the length of the shortest non-singular mixed identity ${\rm PSL}_n(q)$ is bounded by $l$. We generalize the Hull--Osin dichotomy for highly transitive permutation groups to linear groups over finite fields. Finally, we show that the rank limit of ${\rm GL}_n(q)$ for $q$ fixed and $n \to \infty$ is mixed identity free.

math.GR↗

arXiv2023

GAGTA 2023 Problem Session

The conference `Geometric and Asymptotic Group Theory with Applications (GAGTA) 2023: Groups and Dynamics' took place at the Erwin Schrödinger Institute on July 17-21. These are the problems that were proposed during the Problem Session: Residual finiteness growth, Geometric v. random walk boundaries, Stability, Cogrowth, Embedding left orderable groups, Schreier growth gap, $\ell^p$ models for hyperbolic groups, Characterizing hyperbolic groups

math.GR↗

arXiv2023

On the length of non-solutions to equations with constants in some linear groups

We show that for any finite-rank free group $Γ$, any word-equation in one variable of length $n$ with constants in $Γ$ fails to be satisfied by some element of $Γ$ of word-length $O(\log (n))$. By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group $Γ$. Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including $\mathrm{PSL}_d(\mathbb{Z})$ for all $d \geq 2$, and the fundamental groups of all closed hyperbolic surfaces and $3$-manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group $Γ$ and a sequence of word-equations with constants in $Γ$ for which every non-solution in $Γ$ is of word-length strictly greater than logarithmic.

math.GR↗

arXiv2023

The length of mixed identities for finite groups

We prove that there exists a constant $c>0$ such that any finite group having no non-trivial mixed identity of length $\leq c$ is an almost simple group with a simple group of Lie type as its socle. Starting the study of mixed identities for almost simple groups, we obtain results for groups with socle ${\rm PSL}_n(q)$, ${\rm PSp}_{2m}(q)$, ${\rm P Ω}_{2m-1}^\circ(q)$, and ${\rm PSU}_n(q)$ for a prime power $q$. For such groups, we will prove rank-independent bounds for the length of a shortest non-trivial mixed identity, depending only on the field size $q$.

math.GR↗

arXiv2022

Short Laws for Finite Groups of Lie Type

We produce new short laws in two variables valid in finite groups of Lie type. Our result improves upon results of Kozma and the second named author, and is sharp up to logarithmic factors, for all families except possibly the Suzuki groups. We also produce short laws valid for generating pairs and random pairs in finite groups of Lie type, and, conditional on Babai's diameter conjecture, make effective the dependence of our bounds on the rank. Our proof uses, among other tools, the Classification of Finite Simple Groups, Aschbacher's structure theorem for maximal subgroups for classical groups, and upper bounds on the diameters of finite simple groups due to Breuillard, Green, Guralnick, Pyber, Szabo and Tao.

math.GR↗

arXiv2022

Controlling LEF growth in some group extensions

We study the LEF growth function of a finitely generated LEF group $Γ$, which measures the orders of finite groups admitting local embeddings of balls in a word metric on $Γ$. We prove that any sufficiently smooth increasing function between $n!$ and $\exp(\exp(n))$ is close to the LEF growth function of some finitely generated group. This is achieved by estimating the LEF growth of some semidirect products of the form $FSym (Ω) \rtimes Γ$, where $Γ\curvearrowright Ω$ is an appropriate transitive action, and $FSym (Ω)$ is the group of finitely supported permutations of $Ω$. A key tool in the proof is to identify sequences of finitely presented subgroups with short "relative" presentations. In a similar vein we also obtain estimates on the LEF growth of some groups of the form $E_Ω (R) \rtimes Γ$, for $R$ an appropriate unital ring and $E_Ω (R)$ the subgroup of $Aut_R (R[Ω])$ generated by all transvections with respect to basis $Ω$.

math.GR↗

arXiv2022

Quantifying lawlessness in finitely generated groups

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_Γ : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_Γ$ is bounded iff $Γ$ has a nonabelian free subgroup. By contrast we construct, for any nondecreasing unbounded function $f: \mathbb{N} \rightarrow \mathbb{N}$, an elementary amenable lawless groups for which $\mathcal{A}_Γ$ grows more slowly that $f$. We produce torsion lawless groups for which $\mathcal{A}_Γ$ is at least linear using Golod-Shafarevich theory, and give some upper bounds on $\mathcal{A}_Γ$ for Grigorchuk's group and Thompson's group $\mathbf{F}$. We note some connections between $\mathcal{A}_Γ$ and quantitative versions of residual finiteness. Finally, we also describe a function $\mathcal{M}_Γ$ quantifying the property of $Γ$ having no mixed identities, and give bounds for nonabelian free groups. By contrast with $\mathcal{A}_Γ$, there are no groups for which $\mathcal{M}_Γ$ is bounded: we prove a universal lower bound on $\mathcal{M}_Γ(n)$ of the order of $\log (n)$.

math.GR↗

arXiv2021

Quantifying local embeddings into finite groups

We study a function $\mathcal{L}_Γ$ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group $Γ$. We compute this "LEF growth" function in some examples, including certain wreath products. We compare LEF growth with the analogous quantitative version of residual finiteness, and exhibit a family of finitely generated residually finite groups which nevertheless admit many more local embeddings into finite groups than they do finite quotients. Along the way, we give a new proof that B.H. Neumann's continuous family of $2$-generated groups contains no finitely presented group, a result originally due to Baumslag and Miller. We compare $\mathcal{L}_Γ$ with quantitative versions of soficity and other metric approximation properties of groups. Finally, we show that there exists a "universal" function which is an upper bound on the LEF growth of any group on a given number of generators, and that (for non-cyclic groups) any such function is non-computable.

math.GR↗

arXiv2021

Topological full groups of minimal subshifts and quantifying local embeddings into finite groups

We investigate quantitative aspects of the LEF property for subgroups of the topological full group $[[ σ]]$ of a two-sided minimal subshift over a finite alphabet, measured via the LEF growth function. We show that the LEF growth of $[[ σ]]^{\prime}$ may be bounded from above and below in terms of the recurrence function and the complexity function of the subshift, respectively. As an application, we construct groups of previously unseen LEF growth types, and exhibit a continuum of finitely generated LEF groups which may be distinguished from one another by their LEF growth.

math.GR↗