Papers

The determinant approximation conjecture states that for a residually finite group $G$ and a chain $G_i$ of finite index normal subgroups with trivial intersection, the Fuglede-Kadison determinant of every matrix over the group algebra $\mathbb{Q} G$ is the limit of the Fuglede-Kadison determinants of the matrix reductions from $\mathbb{Q} G$ to $\mathbb{Q} (G/G_i)$. We disprove the conjecture by exhibiting an explicit element of the integral group ring of the Heisenberg group with determinant equal to $2$ and reduced determinants bounded by $\sqrt[3]{5}$.

Let $\mathbb O$ be the real octonion division algebra and $\mathfrak g_2=\operatorname{Der}(\mathbb O)$. A quaternionic reflection $T\in G_2=\operatorname{Aut}(\mathbb O)$ fixes a quaternionic subalgebra pointwise and negates its orthogonal complement. For the associated compact symmetric pair $\mathfrak g_2=\mathfrak k_T\oplus\mathfrak m_T$, we give an explicit factored polynomial criterion for generation by an independently chosen even element and odd element. Equivalently, for the reflected pair $(Z,TZT)$ with $Z\in\mathfrak g_2$, the criterion is $\mathcal U_{38}=\mathcal R_{30}\mathcal H_8>0$. The degree-30 Gram determinant detects reducibility on $\operatorname{Im}\mathbb O$, while the degree-8 factor detects generated algebras of dimension at most three. A principal $\mathfrak{su}(2)$ is the remaining irreducible proper possibility. Such a reflected principal closure is realizable for a nonzero $Z$ if and only if its positive rotation frequencies have ratio $1:2:3$; for each such $Z$, all realizing reflections are parametrized by a two-torus and an open interval. A different restriction arises when $Z$ must lie in the stabilizer algebra $\mathfrak k_A\simeq\mathfrak{so}(4)$ of a fixed quaternionic subalgebra $H_A$. For every reflection moving $H_A$, a fixed list of six elements of $\mathfrak k_A$ contains a generating choice; reflections preserving $H_A$ admit none. For one explicit relative configuration $(\mathfrak k_A,S)$, we describe the entire nongenerating set by four geometric families and by six irredundant systems of scalar equations. We determine its real dimension and all generated algebras of dimension at most three; a homogeneous degree-34 polynomial packages the six tests. The degree is not asserted to be canonical or minimal. Two explicit local coordinate maps use short words in a finite-time pulse and one reflection, whose generated subgroup is dense in $G_2$.

We determine the additive structure of the integral cohomology groups with trivial coefficients of the Artin groups in the spherical families $A_n,C_n,D_n$ and the affine families $\widetilde B_n,\widetilde C_n,\widetilde D_n$. We first compute the integral cohomology of braid groups by algebraic methods and then use the Salvetti complexes to express the cohomology of the remaining families in terms of these computations. The resulting descriptions apply in every admissible rank. For each family, we establish cohomological stabilization and show that the integral cohomology groups contain no elements of order $p^2$ for any prime $p$. We also give uniform generating series formulas describing the stable integral cohomology in terms of that of the braid groups.

We establish three results on decision problems for finitely generated matrix groups, based on fiber products: $\bullet$ The torsion-freeness problem is undecidable for subgroups of $\mathrm{SL}_{4}(\mathbb{Z})$, $\bullet$ There exists a finitely generated subgroup $G$ of $\mathrm{SL}_{4}(\mathbb{Z})$ with undecidable square root problem, $\bullet$ No algorithm can, on input a tuple of invertible matrices, output the list of marked finite quotients of the group they generate.

Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by $\mathrm{SSN}_2$, sharply restricts the nonabelian composition factors of the group to the family $\PSL_2(q)$, where $q\geq7$ is a prime power satisfying $q\equiv3\pmod4$. Conversely, this family is precisely the nonabelian finite simple groups that satisfy $\mathrm{SSN}_2$. Consequently, a finite group satisfying $\mathrm{SSN}_2$ is solvable if and only if it has no section isomorphic to one of these groups.

We show that every countable group $G$ that has Ozawa's property $\rm PHP$ and contains a non-abelian free subgroup has the following property: the group ring $KG$ is primitive for any field $K$. As a consequence, we deduce that all non-trivial countable linear groups with trivial amenable radical have this property. Along the way, we prove that the class of groups with $\rm PHP$ enjoys some permanence properties that are of independent interest.

We calculate the quotient of the mod-two cohomology of a product of symmetric and alternating groups by the ideal of negligible elements, namely those that restrict trivially to any order-two subgroup. This quotient arises in the study of cohomological invariants of fields, as by work of Serre the ideal of negligible elements coincides with the ideal of elements that pull back trivially in Galois cohomology. Our calculations answer questions recently posed to us by J-P. Serre, whose recent results we build on. It is a pleasure to share these results on the occasion of his 100th birthday.

A subgroup $H$ of Thompson's group $F$ is closed if every piecewise-$H$ function in $F$ belongs to $H$. Prominent examples of closed subgroups are the maximal subgroups of infinite index, stabilizers and pointwise stabilizers of sets of points, and many of Jones' subgroups. A closed subgroup is finitely generated if and only if its Stallings $2$-core is finite, and it is then isomorphic to a diagram group over the core, in the sense of Guba and Sapir. We study the finiteness properties of closed finitely generated subgroups of $F$ with finitely many orbits on the dyadic rationals. For such a subgroup $H$ the following are equivalent: $H$ is of type $\mathrm{FP}_2$; $H$ is finitely presented; $H$ is of type $F_\infty$; $H$ has finitely many orbits on pairs of dyadic rationals; the action of $H$ on the dyadic rationals is oligomorphic. If moreover the action of $H$ on $(0,1)$ is minimal, these conditions hold if and only if the image of $H$ in the abelianization $\mathbb Z^2$ of $F$ has rank two. All these conditions can be decided from the core of $H$, and when they hold a finite presentation of $H$ can be computed. As a consequence, every finitely generated maximal subgroup of $F$ is of type $F_\infty$.

We prove that the orbifold fundamental group of the moduli space of smooth complex cubic surfaces is a hierarchically hyperbolic group. This places it within a powerful geometric framework whose prototypical examples are mapping class groups. The proof uses a known incomplete complex hyperbolic metric on the moduli space, and applies in a broader setting which includes a number of other moduli spaces.

For a set $W$ of nonempty words, let $S(W)$ be the flat semiring formed by the nonempty factors of words in $W$, together with an absorbing zero. We prove that $S(W)$ has a finite identity basis whenever every word in $W$ is linear, with no restriction on the size of $W$ or on the lengths of its words. In the bounded case its variety is generated by the interval semiring $A_m\cong S(a_1\cdots a_m)$, where $m$ is the maximum word length and the letters $a_i$ are distinct. In the unbounded case its variety is generated by the interval semiring on all finite intervals of the nonnegative integers. We give explicit finite bases in both cases. The proofs encode nonzero polynomial evaluations by endpoint graphs and derive the required graph identifications using finitely many splicing identities. In particular, the eleven-element semiring $S(abcd)$ is finitely based, providing a counterexample to the length-bound conjectures for $S(W)$ proposed by Gao, Ren and Zhao.

We introduce a notion of nilpotency for surjective quandle homomorphisms, which relativizes the nilpotency of quandles recently defined by Darné. A surjective homomorphism is called nilpotent if its relative inner automorphism group is nilpotent relative to the inner automorphism group of the source. We show that this condition can equally be described through the relative transvection group and the relative adjoint group, and that it is equivalent to being a finite composite of covering homomorphisms. Extending the relative lower central series to transfinite ordinals, we then introduce hypocentral homomorphisms and characterize them as the surjections that are right orthogonal to the strongly connected homomorphisms. As a consequence, the strongly connected and the hypocentral homomorphisms form an orthogonal factorization system for the surjections in the category of quandles. Finally, we also introduce reduced and operator-reduced homomorphisms, which relativize reduced quandles, and give sufficient conditions for such a homomorphism to be nilpotent.

For every rational number $\alpha\geq 2$, we construct a 2-step nilpotent group with conjugator length function $\operatorname{CL}(n)\simeq n^\alpha$. We deduce that the nilpotent conjugator length spectrum is dense in $\{0\}\cup[2,\infty)$, even after restricting to groups of nilpotency class at most $2$. Moreover, we show that, despite being a commensurability invariant of nilpotent groups, conjugator length is not a quasi-isometry invariant: for every $m \geq 1$, the cocompact lattices in $\mathsf H(\mathbb R)^m$ realize exactly the growth types $n^2,\ldots,n^{m+1}$. Finally, for every real cubic algebraic number $\theta$, we construct a 2-step nilpotent group such that $n^3 \preceq \operatorname{CL}(n) \preceq n^{3+\varepsilon}$ for every $\varepsilon > 0$, with $n^3 \prec \operatorname{CL}(n)$ if and only if at least one real conjugate of $\theta$ has unbounded partial quotients. Consequently, determining the conjugator length function for arbitrary 2-step nilpotent groups is at least as hard as settling the bounded-partial-quotient problem for real cubic algebraic numbers with two nonreal conjugates.

In this paper we introduce a potentially infinite process of reflections applied to a sequence of points in the dual of the Tits cone associated with an arbitrary finitely generated infinite Coxeter group. This process is infinite whenever the starting point is not in the imaginary cone of the given Coxeter group. We show that this process exhibits well-defined asymptotic behaviour. We study the resulting limits, and give a characterisation of these limits. We then apply the techniques developed from this asymptotic process to the study of infinite sequences of roots totally ordered by the dominance partial ordering.

This article investigates the duality of abelian topological groups by delving into the interplay between the algebraic and topological properties of a group $G$ and those of its dual group. In particular, extending the notion of a $g$-barrelled group, we introduce and study the classes of $\kappa$-barrelled and $\omega$-barrelled groups. Our first main result establishes that every $\omega$-barrelled group is $\aleph_0$-barrelled (i.e., if every weakly convergent sequence in the dual group is equicontinuous, then every metrizable weakly compact subset of the dual group is also equicontinuous). From this, we deduce that the free topological abelian group $A(K)$ over a metrizable compact space $K$ is determined by the subgroup generated by any of its dense subsets. Furthermore, we prove that if $G$ is a $\sigma$-compact, $\omega$-barrelled group, then its dual group $\widehat{G}_\kappa$ equipped with the compact-open topology is sequentially complete (complete if $G$ is hemicompact). We also solve an open question posed by Trigos-Arrieta by constructing an explicit example of an $\omega$-barrelled metrizable group that fails to be $g$-barrelled. Finally, we investigate the dualization of groups lacking infinite compact subsets, proving that for a totally bounded abelian group $G$, every compact subset of $G$ is finite if and only if its dual group $\widehat{G}$, endowed with the finite-open topology, is unordered Baire-like.

Let $G$ be a simple, simply connected algebraic group scheme defined over $\mathbb{F}_{p}$, and let $G_{r}$ be the $r$th Frobenius kernel. Donkin's famous Tilting Module Conjecture purports that a given indecomposable injective $G_{r}$-module can be realized as the restriction of a specific tilting module. The conjecture was first stated in 1990 and withstood proof for nearly 30 years. The authors discovered a counterexample in 2019. This paper provides some indication about the rich mathematics surrounding the Tilting Module Conjecture that also discusses an older conjecture by Humphreys and Verma. New versions of the Tilting Module Conjectures are formulated given multiple families of new counterexamples. Finally, through a new calculation, the authors present further evidence that the Tilting Module Conjecture should hold for reductive algebraic groups with an underlying root system of Type $\mathrm{A}_{n}$ for all fields of characteristic $p > 0$.

We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if $A$ is finite, the ambient group has exponent $r$, and $|A+A|\leq K|A|$, then \[ |\langle A\rangle|\leq r^{(2+o(1))K}|A|. \] This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox–Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent $r$, we also prove that $|A^3|\leq K|A|$ implies $|\langle A\rangle|\leq r^{(2+o_K(1))K}|A|$. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.

For a finite group $G$ let $\omega(G)$ denote its spectrum, the set of orders of its elements, and set $\eta(G)=|\omega(G)|$, the number of distinct element orders. The set $\omega(G)$ has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of $\eta$. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound $1+\sum_{p\mid|G|}v_p(\exp G)\le\eta(G)\le\tau(\exp G)$, in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving $\eta(G)=\tau(\exp G)$. We also prove $\eta(G)\le\tau(|G|)$ with equality only for cyclic groups, and $\eta(G)\ge\pi(|G|)+1$ with equality only when every nontrivial element has prime order. We show that $\eta(G)\le3$ forces solvability by an argument resting only on Burnside's theorem, and, combining this with the recognition of $A_5$ by its spectrum, that $A_5$ is the unique non-solvable group with $\eta(G)=4$. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of $\eta$ over all groups of a fixed order.

Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $\alpha\ge 1$, we construct a finitely generated group $G_{n,\alpha}$ that embeds into the group of $C^\alpha$-diffeomorphisms of $M$, but does not embed into the group of $C^\beta$-diffeomorphisms of $M$ for any $\beta>\alpha$.

The concept of synchronization has been transferred from finite automata, via transformation semigroups, to permutation groups: a permutation group $G$ on a finite set $\Omega$ is said to be synchronizing if, for any non-permutation $t$ on $\Omega$, the transformation monoid $\langle G,t\rangle$ contains an element of rank~$1$. Synchronizing permutation groups are primitive and basic, and so are affine, diagonal, or almost simple (in the O'Nan–Scott classification). In an earlier paper, the author, with João Araújo and Ben Steinberg, surveyed what was known at the time about synchronizing permutation groups. Since then, several new classes of groups have been tested for the synchronizing property, using the fact that a group is non-synchronizing if and only if it preserves a nontrivial weakly perfect graph. Cases studied include diagonal groups with more than two socle factors, two natural actions of symmetric groups, and some actions of classical groups; the proofs are scattered in the literature. My primary purpose is to collect these results with their proofs. As a second objective, I will explain the background, and comment on some related topics. (Reading this paper does not require knowledge of the earlier paper.)

A large $p$-subgroup of a group $G$ is a self-centralizing $p$-subgroup $Q \le G$ whose normalizer controls the normalizers of all the non-trivial, central subgroups of $Q$. In 2016 Meierfrankenfeld, Stellmacher and Stroth produced a result describing the $p$-local structure of a finite group having a large $p$-subgroup (the main examples arising from finite groups of Lie type in defining characteristic $p$). This result is a major success within the wider framework of studying groups of local characteristic $p$. We attempt to produce a result analogous to that of Meierfrankenfeld, Stellmacher and Stroth, but for fusion systems and localities. Previous work of Ellen Henke and the author shows that reasonable generalizations can be formulated and solved equivalently either within the realm of fusion systems or in the world of localities. In particular, in the present paper we set the stage for our analysis, showing how working within a locality grants a clear advantage: it allows to follow the same lines of reasoning as for a group. We therefore produce analogous reduction results and case subdivision as those in the 2016 result of Meierfrankenfeld et al. and, proceeding according to the analogy, we deal with occurrences of certain natural orthogonal modules and with the first of the cases that are to be studied, the so-called symmetric case. The remaining cases will appear in future publications.

Let $H$ be the generic Hecke algebra over $\mathbb{Z}[\mathbf{q}^{\pm 1}]$ associated to a split Kac–Moody group $G$, arising as the deformation of the group algebra of its Weyl group $W$. The cocenter $\overline{H} = H/[H, H]$ encodes the trace and character theory of $H$, playing a fundamental role in representation theory and harmonic analysis. Through deep combinatorial results on cyclic reductions in Coxeter groups, each conjugacy class $\mathcal{O}$ of $W$ determines a canonical element $T_{\mathcal{O}}$ in $\overline{H}$, and these elements are known to span the cocenter. However, establishing their linear independence has remained an open problem outside of finite and affine types. In this paper, we solve this problem: the canonical elements form a $\mathbb{Z}[\mathbf{q}^{\pm 1}]$-basis of the cocenter $\overline{H}$. Our approach differs from earlier representation-theoretic methods in finite and affine types. To construct explicit functionals that separate all conjugacy classes, we develop a new framework based on re-normalized orbital integrals. This framework synthesizes parabolic induction, traces of infinite-dimensional bimodules, and Kac–Moody harmonic analysis into an almost-dual basis for the cocenter. As a key local ingredient, we establish a generic duality theorem for finite groups of Lie type relating the cocenter to regular semisimple conjugacy classes, and determine precisely when this pairing is non-degenerate. Finally, we deduce the existence and uniqueness of generic class polynomials for $W$, and prove a uniform “dimension=degree” theorem for basic Deligne–Lusztig varieties of the split Kac–Moody group $G$.

In a 2023 survey paper, Gabriel Navarro posed the following problem: Given an irreducible character $\chi$ of some solvable group $G$ where $\chi$ either is 2-rational or has odd degree, does there exist some $g \in G$ such that $\mathbb{Q}(\chi(g)) = \mathbb{Q}(\chi)$? The answer was recently shown to be no in general by Ulrich Thiel. However, when $\chi$ is further assumed to be factorizable as a product of $p$-special characters, we are able to show that the answer is yes.

Let $\Gamma$ be the Cayley graph of Thompson's group $F$ with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density $7/2$, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with \[ \operatorname{dens}(\Gamma)>3.50074529. \] The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed $a\ge0$, the optimal limiting density of a rule depending on the window $[-a,1]$ is exactly $7/2$. Thus arbitrary finite left context together with one right-hand category does not improve the Belk–Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of $\mathbb Q(\sqrt3,\sqrt{2\sqrt3-1})$.

We give a general method for proving Kazhdan's property $(T)$ for quotients $G/K_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G/[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients $\operatorname{Mod}(\Sigma_g)/(\mathcal{T}_g)_{[c+1]}$ for $g\ge3$ we show that they have property $(T)$ for every $c\ge1$. We also prove property $(T)$ for $\operatorname{Aut}(F_3)/(\mathrm{IA}_3)_{[c+1]}$ for every $c\ge1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for $c\ge2$ are not residually finite, and thus non-linear.

For any infinite cardinal $\kappa$, we give an example of a centerless group $G$ of cardinality $\kappa$ whose automorphism tower terminates after $\kappa^+$ steps, answering a question of Simon Thomas. The work on this paper was assisted by AI.

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot–Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.

We construct a $3$-generator $9$-relator group with unsolvable word problem. We use the group to construct two fixed-size Adian–Rabin families of group presentations, one with $4$ generators and $11$ relators, and another with $2$ generators and $10$ relators. As a consequence, $\#_7(S^2\times S^2)$ is topologically unrecognizable and $\#_9(S^2\times S^2)$ is smoothly unrecognizable. These algebraic and topological results improve the previous best known bounds by Borisov, Tancer, and Gordon. The construction of the group builds upon an example of Borisov and uses additional HNN extensions and Tietze eliminations to reduce the size of the presentation. We also provide a machine-checked Lean~4 formalization of the algebraic results.

We propose to define amenability of a Lie algebra by the existence of almost-invariant finite-dimensional subcoalgebras of its universal enveloping algebra. More generally, a module coalgebra of a cocommutative Hopf algebra is amenable if it admits almost-invariant finite-dimensional subcoalgebras. We prove a coalgebraic rounding theorem: almost-invariant finite-dimensional subspaces can be replaced, without loss in the Følner constant, by almost-invariant finite-dimensional subcoalgebras. For Lie algebras this implies that our definition is equivalent to Elek's amenability of the left regular module of the universal enveloping algebra seen merely as an associative algebra. For groups this recovers the result that a group is amenable if and only if its group ring is algebraically amenable. It furthermore shows that, for every amenable group, all its nonzero modules are amenable, thus proving an assertion by Gromov. We prove that amenable Hopf algebras are closed under taking subalgebras, quotients, cleft extensions, and directed unions, and that every Hopf algebra locally of subexponential growth is amenable. We give examples of amenable Lie algebras which are not elementarily amenable. Finally, we show that amenability passes to the associated graded Hopf-module coalgebra.