In mathematics, a group is not a crowd on a street corner so much as a rulebook for combining objects. Some rulebooks are brutally simple: add, multiply, repeat, and you’re done. Others are more subtle, their implications rippling through the entire structure. A “law” in this world is a nontrivial word in a free group that, no matter how you map the word’s letters into a particular group, always lands at the identity. If a group contains such universal laws, it’s, in a sense, easier to pin down. Bradford and Willis flip that intuition on its head: what if a group has zero universal laws at all, and we want to measure how “wild” that lawlessness actually is? They don’t just ask the question; they develop a precise, searchable way to tune how wild it can get. The result is a kind of dial for mathematical chaos, one that can be turned to match almost any unbounded growth pattern you can imagine.
To a general reader, this sounds abstract, even arcane. But the core ambition is familiar: you want to know how far reality can stretch before it snaps back into a predictable pattern. Lawlessness growth is a quantitative way to track that stretch. If a group is lawless, its lawlessness growth function never collapses to a neat bound; the longer you push the word length, the more opportunities you have to witness nontrivial behavior. The slower this growth, the stronger the sense that the group’s wildness is somehow more constrained. Bradford and Willis show you can craft groups whose lawlessness growth matches almost any reasonable target function, all while staying inside a family of groups that are, by many standards, “tame.” The paper is a workshop on control: not the control of chaos itself, but the control of chaos’s expression in the world of groups.
The study behind these ideas comes from Henry Bradford and Jacob Willis, researchers who published their arXiv preprint in March 2025. Working in the sphere of group theory, they construct something they call sparse wreath products and then demonstrate that, by tweaking a handful of ingredients, they can align the lawlessness growth of a finite-generated group with essentially any unbounded nondecreasing function. In other words, they’re not just proving existence of wild groups; they’re giving you a family portrait where you can choose the growth pattern and watch the group bend to it. The results sit at the intersection of two big currents in modern algebra: building new objects by gluing together simpler pieces, and grappling with how far an infinitely large, yet finitely generated, object can stray from familiar laws.
Bradford and Willis’ work focuses on a particular lane within the vast universe of groups: elementary amenable groups. These are built from simple building blocks like finite groups and the integers, closed under a few natural constructions. They’re sort of the “friendly neighborhood” among infinite groups—where you might hope to see some order, some predictability. And yet, within that neighborhood, the authors show you can thread a needle that generates lawless behavior of astonishing variety. The punchline is both mathematical and philosophical: even among groups that feel tame, the wildness can be engineered with surprising precision. The authors’ commitment to a transparent construction—sparse wreath products—gives other researchers a clear blueprint to test, extend, or repurpose in future work.
To ground the story in a concrete sense of authorship and place, this study comes from Bradford and Willis, researchers whose affiliations are listed in the preprint. The paper emphasizes their shared belief that a carefully engineered “sparseness” can isolate the behavior of a group at different scales, letting the authors tune how often a nontrivial word fails to vanish. It’s a delicate balance: keep the whole construction connected, keep it amenable, and still allow a staggering amount of freedom in how the lawlessness grows. The result is a set of theorems that reads like a menu for building lawless groups with particular long-run properties—an achievement that feels less about a single exotic object and more about a reliable method for sculpting infinite algebraic worlds.
What does lawlessness mean in mathematics
At its core, a “law” in group theory is a word in the free group that vanishes under every homomorphism into a fixed target group Γ. If you’re willing to map the word to every possible substitution of the target’s elements, and you always end up at the identity, that word is a law for Γ. A group that has no such universal law—no matter how you press the words into Γ— is lawless. It’s the algebraic analogue of a city where no single rule governs every street corner; you can always find a different local rule lurking just around the next bend.
To quantify this wildly conceptual idea, Bradford and Willis introduce the lawlessness growth function. Imagine you take all nontrivial words of length at most n in a two-generator free group, and for each, you ask: can I find a short enough tuple of group elements whose evaluation of that word is not the identity? The minimal length of such a witness, taken over all nontrivial words of length ≤ n, is the value Γ(n) (with choices of generator set kept in mind). If Γ(n) stays small, you can already see a tendency toward zero or near-identity behavior; if Γ(n) grows without bound, you’re traversing deeper into the realm where lawlessness remains, even as you demand longer and longer words be witnessed as failing to vanish.
One of the paper’s clean through-lines is the relationship between lawlessness and more familiar group-theoretic properties. It’s known, for instance, that a group whose lawlessness function is bounded must contain a nonabelian free subgroup. In the other direction, the more slowly the lawlessness grows, the stronger the sense that the group’s wildness resists being pinned down by any single universal equation. Bradford and Willis push this intuition into a formal statement: they demonstrate that, by their sparse wreath product method, almost any unbounded nondecreasing function can be realized as the lawlessness growth (subject to mild growth conditions). It’s a kind of “inverse problem” in which you specify the growth you want and build a group that achieves it.
Sparse wreath products and the art of separation
To build these tailor-made lawless groups, the authors invent and refine a construction they call sparse wreath products. The starting point is a base group Δ and the unrestricted wreath product with the integers, Δ Wr Z. In a wreath product you imagine many copies of Δ lined up along the integer line, with the Z-part acting by shifting which copy you’re looking at. It’s a classic device in group theory for producing new, interesting symmetries from simpler movers.
What makes Bradford and Willis’ version “sparse” is a precise limitation: they pick a finite set S of generators in Δ^Z (the direct product of copies of Δ indexed by the integers) and require that translates of the supports of these generators intersect at most at a single point in Z. In everyday language, the different pieces of the puzzle barely talk to one another; each piece acts mostly in its own little neighborhood, with only a whisper of interaction elsewhere. That sparsity is the secret ingredient. It lets the authors claim that the way words evaluate on the whole group mirrors the way words evaluate on the individual Δ blocks, at scales we can control by choosing outputs of certain functions that guide how the pieces are put together.
Crucially, these sparse wreath products stay within the realm of elementary amenable groups. That is, they do not suddenly acquire free subgroups of the kind that typically explode the complexity of a group’s geometry. This is not just a cosmetic constraint. It means you can have incredibly nuanced behavior—the presence or absence of universal laws—without stepping outside a class that mathematicians often consider tractable and well-behaved. The sparseness thus becomes a robust engine for constructing lawless groups while staying in a safe neighborhood of group theory where many tools still apply.
There’s a second, more technical payoff to the sparseness: the authors can prove that the constructed groups are not residually finite. A residually finite group is one that looks “finite” from every angle: for any nontrivial element, there’s a finite quotient in which that element remains nontrivial. The paper shows that, under their setup, the resulting sparse wreath product cannot be approximated by finite quotients. This is a striking reminder that even very tame-looking infinite groups can resist finite-approximation tricks, a theme that resonates through many corners of modern algebra.
Can we prescribe growth? The results and their scope
The heart of Bradford and Willis’ contribution is the remarkable claim that you can tune the lawlessness growth function to track almost any unbounded nondecreasing function f. The theorems come with a few mild technical caveats, but the spirit is simple: given f, there exists an elementary amenable lawless group Γ with a finite generating set S such that for every natural number n, the lawlessness growth Γ(n) is comparable to f(n). In other words, you can realize f as the growth pattern of a lawless group within a controlled, amenable universe.
How do they build such a creature? A key technical ingredient is a careful arithmetic scaffolding using PSL2(p) groups, the two-by-two projective special linear groups over finite fields. Bradford and Willis show that for a prime p, the shortest law in PSL2(p) W_p (the wreath product piece tied to p) has length roughly between (p−1)/3 and 8p+6. That provides a controlled supply of short ”laws” to seed the construction. Using primes p_n chosen in relation to a nondecreasing function L(n), they assemble a direct sum of PSL2(p_n) W_n blocks, each contributing a law of a controlled length, and they then weave these blocks together inside the sparse wreath product in a way that the overall growth of lawlessness depends on the chosen functions p, q, and L. The upshot is a precise recipe: zoom in on scale n, pick a prime and a block with a law of a known short length, and let the global construction force the lawlessness witness to grow according to the target f.
From there the authors prove two complementary directions. First, they show that for any unbounded f satisfying a modest growth condition, there exists a group whose lawlessness growth is not just large but matches f up to the usual equivalence of growth rates. Second, when you pair f with another function g that acts like a stable, identity-tracking baseline, you can ensure that the lawlessness growth sits as a precise equivalent to f(n) — no hidden slack, no creeping deviations. They push this through a sequence of carefully engineered lemmas (in particular around the behavior of commutators and the supports of elements in the sparse wreath product), culminating in a robust toolkit for sculpting a group’s lawlessness to order.
What’s particularly striking is that this dial can be set for a broad spectrum of shapes. The authors point to explicit families, such as f(n) = ceil(n^λ) for any λ ≥ 1, showing you can realize polynomial growth rates with the same machinery that yields more exotic growth—subexponential, superpolynomial, or anything in between—within the same conceptual framework. They also outline a path to even wilder patterns by pairing their L, p, q machinery with a second mechanism that injects a different growth driver, all while remaining in the safe harbor of elementary amenability.
In addition to the positive construction results, the paper also clarifies limitations of their method. Notably, the groups produced by their core construction are not residually finite. This is not a defect so much as an intrinsic feature: the very way the pieces talk to one another—sparseness at scale—produces a global object that cannot be detected by peering through finite quotients. This aligns with a broader sense in group theory that sometimes the wildest algebraic behavior refuses to be captured by finite shadows, even when the group itself wears a friendly face.
Why this matters: from theory to bigger questions
At first glance, the result might feel like a boast about mathematical craftsmanship: you can build a zoo of lawless groups and tune how lawless grows. But there are deeper currents here. One is a challenge to a long-standing intuition that wild algebraic behavior must come with some heavy, nonamenable baggage. Bradford and Willis show that tame, amenable scaffolds can host an astonishing variety of lawlessness. The second is a reminder that complexity can be engineered with a surprising degree of control. The sparse wreath product is not just an abstract gadget; it’s a blueprint for turning qualitative ideas—laws, lawlessness, growth—into quantitative, comparable objects that can be studied side by side, like species in a well-curated ecological exhibit.
These results also speak to a perennial question in mathematics: how far can a single concept be stretched before the structure ceases to behave in predictable ways? By showing that lawlessness growth can be tailored to match a wide range of growth curves, Bradford and Willis reveal a spectrum of possible behaviors that was previously hidden behind a veil of general theorems. It’s a reminder that even within a well-ordered universe, you can have precisely the kind of flexibility that makes a field exciting—where the same basic definitions yield radically different and tunable futures depending on how you assemble the pieces.
There are also connections, however indirect, to other themes in contemporary mathematics. The notion of residually finite groups—those that can be approximated by finite quotients—rings bells in computational group theory, topology, and logic. Demonstrating not-residually-finite behavior in their constructed objects puts a spotlight on the limits of finite methods for understanding infinite groups. And by anchoring their construction in PSL2(p) blocks, the authors connect to the rich world of finite simple groups, where short identities and laws often play outsized roles in the structure theory that follows them.
Beyond pure theory, the paper invites a broader reflection on how we talk about “wildness” in mathematics. If you can prescribe the growth of a group’s lawlessness, what does that say about how we measure complexity? Could similar ideas be transported to other algebraic or combinatorial universes—trees, automata, or even networked systems—where the notion of a universal constraint (a law) might or might not exist? Bradford and Willis don’t pretend to have answered those questions, but they light a path toward them, a way to speak about abstraction with the precision of an engineer calibrating a device.
For readers who love the thrill of a clever construction, the sparse wreath product is a quiet revelation: a compact idea that unlocks a vast landscape. It’s a reminder that in mathematics, control and creativity can walk hand in hand, and that even within the gentlest corners of the subject, one can choreograph a symphony of possible futures. Bradford and Willis offer not just a theorem, but a framework—a toolkit for anyone who wants to imagine a world of groups where laws are scarce, but the rules for building them are abundant.
As with many frontier works, the authors leave open questions that invite others to join the conversation. One natural thread is the search for residually finite lawless groups with prescribed growth—does such a creature exist, and if so, how would you coax it to behave? Another is to push the sparse wreath product idea into new families of base groups, to see what other exotic features might emerge when sparseness meets algebraic depth. And, of course, the possibility of applying these ideas to computational questions—how one tests the lawlessness of a group in practice, or whether certain growth profiles can be detected efficiently—remains a tantalizing area for future work.
In short, Bradford and Willis give us a manual for shaping the wildness of math objects at will, while staying within a neighborhood of calm, well-understood groups. It’s a playful, serious contribution that nudges us to see lawlessness not as an unruly anomaly, but as a spectrum that we can explore, chart, and perhaps even harness.
The paper’s invitation to experiment with infinite algebraic worlds is clear: take a basic, familiar construct, add a precise kind of sparseness, and you can orchestrate a whole orchestra of growth patterns. If you’re the kind of reader who loves a blueprint hidden in dense symbols, this work offers a rare payoff—a way to imagine, and then build, new landscapes where rules are scarce, but the rules for building them are plenty.
