Papers

We introduce, for an embedded Legendrian link~$L$ in a contact 3-manifold~$(M,\xi)$, the group of its Legendrian motions. Analogously with the smooth case, we prove that this Legendrian motion group is isomorphic to the fundamental group of the space of unparametrised Legendrian embeddings. For all Legendrian knots in the standard contact 3-sphere, we characterise Legendrian motions in terms of contact mapping classes of the knot exterior. In an appendix, we give a short exposition of our techniques applied to the smooth case, comparing to the classical Dahm homomorphism.

We show some sufficient conditions ( in terms of properties of $A$, $B$ and $X$) for the non-abelian exterior square $G \wedge G$ to be residually finite, where $G $ is the wreath product $A \wr B$ or the combinatorial wreath product $G = A \wr_X B$, in the latter case $B$ is a free group. We prove that for the Grigorchuk group $G$ the non-abelian exterior square $G \wedge G$ and the non-abelian tensor product $G \otimes G$ are residually finite. We show conditions that imply that $A \wr B$ is not (cohomologically) good in dimension $\leq 2$.

For a finite group \(G\), let \(N(G)\) be the simplicial complex of subsets that do not generate \(G\). We separate a general Frattini-topological reduction from the structure special to finite \(p\)-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the order complex of the proper part of the subgroup lattice of \(G/\Phi(G)\). For a finite \(p\)-group this quotient is an elementary abelian vector space, and the core becomes the non-spanning complex of a uniform parallel extension of \(\mathrm{PG}(r-1,p)\), where \(r=d(G)\) and \(q=|\Phi(G)|\). We use this geometry to determine the exact simplicial isomorphism data and full simplicial automorphism group, identify top homology equivariantly with the appropriate restriction of the Steinberg module, and derive modular consequences. We also give an explicit supportwise formula for every multigraded Betti number and compare it with a closed single-sum formula for the \(\mathbb Z\)-graded Betti numbers. Known matroidal and building-theoretic inputs are stated separately from the group-specific consequences. The resulting framework also yields the homotopy type, depth, regularity, projective dimension, face enumeration, and the complete Cohen–Macaulay classification.

We disprove the classical version of the Borsuk-Ulam-flavored conjecture, due to Baum-Dąbrowski-Hajac, that free actions $X\times \mathbb{G}\to X$ on non-empty compact Hausdorff spaces by non-trivial compact groups admit no equivariant maps $X*\mathbb{G}\to X$. Specifically, among non-trivial compact abelian $\mathbb{G}$, the conjecture fails precisely when $\mathbb{G}$ is torsion-free profinite.

A negative answer to Whitehead's asphericity problem would follow from an infinite chain of two-complexes, starting with a non-aspherical one, in which every inclusion induces the zero map on second homotopy groups. We prove that a connected two-complex $K$ is the first term of such chains of every finite length if and only if $K$ has a connected acyclic regular cover. In particular, there is a finite non-aspherical two-complex $K$, a presentation complex of $\mathrm{SL}(2,5)$, such that for every $n$ there is a strictly increasing chain of finite two-complexes $K=K_0\subset K_1\subset\dots\subset K_n$ in which all inclusions are zero on $\pi_2$.

We define coarse variants of the notions of connectivity, hamiltonicity, toughness, and spanning trees. Our notions are invariant under quasi-isometries, and they obey the same relationships between them as their classical analogs. We pose several open problems that are coarse versions of classical results and conjectures about hamiltonicity.

Let $\mathcal{G}$ be a finite connected simple graph with at least two vertices. We construct a finite regular hypertope $\Gamma$ whose diagram is the graph $\mathcal{G}$ with every edge labelled by $4$, and for which $Cor(\Gamma)/Aut(\Gamma) \cong Aut(\mathcal{G})$. This shows that any possible group of diagram symmetries can be realized by the correlations of a hypertope. The construction uses a group generated by involutions, with nilpotency class two, exponent four and commutator relations dictated by the defining graph.

We study the $\ell^2$-Betti numbers of graph products of cyclic groups. We relate these invariants to the $\ell^2$-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first $\ell^2$-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first $\ell^2$-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.

We give an algebraic criterion for certain sequences $(G_n)_{n\in\mathbb{N}}$ of groups to have Kazhdan's property $(T)$ eventually. As a consequence, we prove that the Brin–Thompson groups $nV$ have Kazhdan's property $(T)$ for large $n$. The algebraic criterion provides a uniform proof that $nV$, $\mathrm{Aut}(F_n)$ and $\mathrm{EL}_n(R)$ for a finitely generated associative unital ring $R$ have property $(T)$ for large $n$. In an appendix, written by Francesco Fournier-Facio, it is shown that the group $nV$ for large $n$ is neither hyperlinear, in particular not sofic, nor $\mathrm{MF}$. Further, it is not Schatten $p$-approximable for any $1\le p<\infty$.

Motivated by an observation concerning the $S_3$-conjecture, we prove that every abelian composition factor of a finite rational group has order $2$, $3$, or $5$. Thompson previously restricted these orders to $\{2,3,5,7,11\}$; we exclude $7$ and $11$, establishing the bound that he proposed. Together with the work of Feit and Seitz on nonabelian composition factors, this completes the determination of the possible composition factors of finite rational groups.

We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.

For a symmetric subset of a group containing the identity, its universal local presentation records the multiplication relations visible within the subset. We prove that a group is recovered from every such generating subset if and only if it is trivial or cyclic of order two or three, as conjectured by P. A. Smith. The main step is an exact completion theorem: local maps into a target group always extend to generating domains if and only if every target element has a square root. By embedding an arbitrary group in a group with surjective squaring, we deduce that every partial local presentation embeds in a generating one. This gives subgroup inheritance and reduces the classification to finite generation. We also construct obstructions in one fixed countable metabelian target on domains of at most five elements. This bound is sharp even when the target is allowed to vary.

It is shown that the mapping class group of any non-sporadic surface admits a continuum of equivariant coarse median structures, thus improving a previous theorem of the author. The same holds for a wide class of right-angled Artin and Coxeter groups, including examples of any dimension at least two, and RACGs which are not quasi-isometric to RAAGs; the coarse median structures in these cases are not induced by cocompact cubulations. The results are instances of a general procedure, which uses quasicocycles to flexibly modify the structure of a (suitable) hierarchically hyperbolic group.

We prove that every countable Borel equivalence relation containing a finite index Borel treeable subequivalence relation is Borel treeable. This answers a long-standing question of Jackson–Kechris–Louveau. Our proof is based on a median subalgebra filtration construction on products of trees, together with an extension of the median graph treeing algorithm of Chen–Poulin–Tao–Tserunyan to certain well-behaved increasing unions.

Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\le M$, and that equality holds only for abelian groups if $|G|$ is odd. We prove that $|G:G'|\le M/p$, unless $p=3$ and $G$ is the direct product of an abelian group and a group isomorphic to $\Gamma(2^3)$ or to $\Gamma(2^3)\times C_3$ acting on $V$ in a specific way, in which case $|G:G'|=3M/7$. In particular, $|G:G'|\le 3M/7$ for every nonabelian group of odd order. If $|V|$ is odd, then $|G:G'|=M/p$ holds if and only if $|G'|=p$; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy $|G:G'|\le 3M/13$, and even $|G:G'|<M/(2p)$ if $p\ge5$. The bound $3M/13$ is best possible.

We construct non-co-sofic invariant random subgroups in connected simple real Lie groups of rank one with finite center, in Zariski connected simple rank-one groups over non-Archimedean local fields, and in Aut(T) for regular and bi-regular trees T. This gives a negative answer to the continuous analogue of the classical Aldous-Lyons conjecture for these groups. Our constructions build on the existence of a finitely generated non-sofic group and answer a question raised in the author's ICM paper [Gel18].

Given a cycle type, the corresponding conjugacy class of $S_n$ generates either $S_n$ or $A_n$ for sufficiently large $n$. We prove that the sequence of diameters of Cayley graphs is eventually polynomial on residue classes for any fixed cycle type. This result is also extended to finite unions of conjugacy classes.

We study isomorphism decision and basis construction for finite Abelian groups of known order $n$. We measure complexity by the number of additions performed in the groups and by the total running time. We give a randomized isomorphism decision algorithm that performs $\tilde O(n^{1/4})$ group additions and runs in $\tilde O(n^{1/4})$ time. This improves the $\tilde O(\sqrt n)$ upper bound of Chen and Fu and matches Bshouty $\Omega(n^{1/4})$ lower bound up to polylogarithmic factors. We also prove that finding a basis with success probability at least $2/3$ requires $\Omega(\sqrt n)$ group additions in the worst case. This matches the $\tilde O(\sqrt n)$ upper bound of Chen and Fu up to polylogarithmic factors. These results separate isomorphism decision from basis construction in finite Abelian groups: their optimal worst-case complexities are $\tilde\Theta(n^{1/4})$ and $\tilde\Theta(\sqrt n)$, respectivel

We prove that deciding whether there is a surjective homomorphism from an arbitrary finitely presented inverse monoid onto the bicyclic monoid is $\mathsf{NP}$-complete. As part of the proof, we show that an extension of existential Presburger arithmetic which involves greatest common divisors on $n$ arguments is in $\mathsf{NP}$, extending a recent result of Défossez, Haase, Mansutti, and Pérez (SODA 2024).

We prove that the centered Hardy–Littlewood maximal operator is not of weak type $(p,p)$ for any $1\le p<\infty$ in three solvable settings: the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ with the switch–walk–switch word metric, the Baumslag–Solitar groups $BS(1,n)$, $n\ge2$, with their digit word metrics, and the three-dimensional Sol group with any left-invariant Riemannian metric and its Riemannian volume. To the best of our knowledge, we give the first examples of finitely generated groups, equipped with word metrics and counting measure, for which the centered Hardy–Littlewood maximal operator fails to be of weak type $(1,1)$.

We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra $\mathcal O_\infty$. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of $\mathcal O_\infty$: one family being in an obvious bijection with the unit sphere of Hilbert's space.

The stylic monoid $\mathrm{Styl}_n$, introduced by Abram and Reutenauer, is the quotient of the plactic monoid by the relations $x^2=x$, and its elements are represented by $N$-tableaux. Volkov showed that the Catalan monoid $\mathrm{Cat}_n$ of order-preserving, order-decreasing self-maps of $\{0,1,\ldots,n\}$ is a quotient of $\mathrm{Styl}_n$. However, the quotient map is defined on generators, and it is not apparent how to see, from an $N$-tableau, the map in $\mathrm{Cat}_n$ it corresponds to. In this paper we give a simple visual way to read off this map, and some of its main properties, from the $N$-tableau. The new ingredient is that we do not insist on drawing an $N$-tableau as a classical Young tableau: we allow the entries of each row to be shifted relative to the row below, as long as each entry stays above a smaller one. We call this a positioning, and prove that the column word read from any positioning is plactically equivalent to the usual column word; so every positioning can be used to compute the quotient map. We work with the tight positioning, in which each entry is pushed as far right as possible, and define the full core of an $N$-tableau: the part of each column that climbs by consecutive values from the bottom row. We call an $N$-tableau full if it equals its full core. We prove that passing to the full core does not change the image in $\mathrm{Cat}_n$, that full $N$-tableaux are in bijection with $\mathrm{Cat}_n$, and we show how to read the corresponding map directly off a full $N$-tableau.

Given a topological space $X$ and a family of subgroups $\mathcal{F}$ of $\pi_1(X)$, the $\mathcal{F}$-category of $X$ is given as one less than the cardinality of the smallest cover of $X$ by open subsets whose fundamental groups lie in $\mathcal{F}$. In this article we study $\mathcal{F}$-categories of aspherical spaces and obtain cohomological lower bounds, a maximality result and bounded cohomology classes whose vanishing properties are crucial for determining $\mathcal{F}$-categories. For this purpose, we study the bounded Adamson cohomology of a family of subgroups and discuss universality properties of bounded cohomology classes. We apply our techniques to derive some applications to the study of the monotonicity of $\mathcal{F}$-categories under degree-one maps between manifolds and to provide a counterexample to a question of Capovilla, Löh and Moraschini.

We formulate a root-conjugation criterion for producing generating pairs and detecting Nielsen inequivalence by a polynomial symmetric-trace invariant. The method simultaneously treats mapping class groups and automorphism groups of free groups. We prove that $\mathrm{Mod}(\Sigma_g)$, $g\ge2$, and $\mathrm{Aut}(F_n)$, $\mathrm{SAut}(F_n)$, $\mathrm{Out}(F_n)$ and $\mathrm{SOut}(F_n)$, $n\ge3$, have countably infinitely many Nielsen classes of generating pairs. A sign parameter unifies the full and special automorphism-group constructions, with common generation arguments and trace polynomials. The same homology representations give the corresponding results for $\mathrm{Sp}_{2g}(\mathbb{Z})$, $g\ge2$, and $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{GL}_n(\mathbb{Z})$, $n\ge3$. This answers Hirose–Monden's low-genus question and Kapovich's Question 8.1.

The Weak Commutativity Group was defined by the first author in the early 1980's as $ \chi(H) = \langle H, H^\psi \mid [h,h^\psi] = 1, \ \forall h \in H \rangle $ where $H$ and $H^\psi$ are isomorphic groups under an isomorphism $\psi$, and proved that $\chi(-)$, seen as a group operator, preserved the classes of solvable groups, finite groups, finite $p$-groups, and perfect groups. The list has grown considerably since then (see references). It is clear that the iterated operator $\chi^n (-)$ also preserves the same classes. However, we observe that for $n \geq 2$ many of the $2^n$ $H$-generating copies of $\chi^n (H)$ fail to weakly commute among themselves. We introduce in this paper the "$n$-ary Weak Commutativity Group" generated by $n$ copies of $H$ which weakly commute among themselves; that is $ \chi_n (H) = \langle H, H^\psi, \dots, H^{\psi_{n-1}} \mid [h^{\psi_i}, h^{\psi_j}] = 1, \ \forall h \in H, \ 0 \leq i < j \leq n-1 \rangle $ (by definition, $H^{\psi_0} = H, H^{\psi_1} = H^\psi$). We study quotients of $\chi^n (H)$ which are also simultaneously quotients of $\chi_{2^n} (H)$; we refer to them as Hybrid $\chi^n (H)$ - $\chi_{2^n} (H)$ groups. For $n=2$ we determine the largest common quotient $Q(H)$ of $\chi^2(H)$ and $\chi_4(H)$, that is, the largest hybrid $\chi^2(H)$-$\chi_4(H)$ group. A related class of groups in our study is $\chi(n,H)$, a quotient of $\chi_n(H)$, previously introduced by the third named author in his 2007 doctoral thesis. We prove that this class of groups continues to preserve finiteness, nilpotency and perfectness, and we provide detailed information about their structure. In contrast, $\chi_n(H)$ behaves differently: $\chi_3(C_2 \times C_2)$ is infinite, while $\chi_3(C_{p^t} \times C_{p^u})$ is finite for $p$ odd.

We construct a deterministic Farey-boundary algorithm converting a cyclic prefix table for an element of $\mathsf T$ into a word in addressed boundary flips. The reduction uses rational boundary pairs and cyclic order, with a marked edge determining the residual modular correction. Its unmarked stage uses at most twice the minimum number of diagonal exchanges. For input size $s$, the complete output has $O(s)$ factors, encoded length $O(s^2)$, and running time $O(s^3)$. Adjoining shuffles extends the construction to $\mathsf V$ with polynomial bounds. We also give explicit boundary generating sets and their local flip–shuffle relations. Changes of generators in the Lochak–Schneps and Bleak–Quick presentations yield a two-flip presentation of $\mathsf T$ and a presentation of $\mathsf V$ with two flips, one shuffle, and nine relations.

We introduce and investigate the irreducibility of endomorphisms of finitely generated free semigroups, i.e., we investigate when an endomorphism $\phi: \Sigma^+ -> \Sigma^+$, where $\Sigma$ is any alphabet, can be nontrivially expressed as a composition $\phi = \psi_2 \circ \psi_1$ of endomorphisms $\psi_1, \psi_2: \Sigma^+ -> \Sigma^+$. We, hence, study a notion of primality in the endomorphism monoid of the free semigroup—a natural and fundamental concept in this algebraic structure. We establish that irreducibility is a nontrivial property for the class of so-called rank-preserving endomorphisms, and we provide a characteristic condition separating the reducible and irreducible endomorphisms. We also characterise when an endomorphism is a factor of another endomorphism, analyse the non-uniqueness of factorisations of a rank-preserving endomorphism into its irreducible components, and investigate the use of incidence matrices to give insights into the (ir-)reducibility of rank-preserving endomorphisms.

The Clifford hierarchy plays a central role in fault-tolerant quantum computation, but its higher levels remain only partially understood. Recent work on quantum higher order Fourier analysis characterizes membership in the Clifford hierarchy using quantum derivatives, but does not provide an explicit description of the individual levels. In this work, we address this problem by showing that the collection of quantum derivatives of a unitary operator forms a non-abelian \(1\)-cocycle, and conversely that every such \(1\)-cocycle arises from a unitary operator. This yields a recursive cohomological characterization of the Clifford hierarchy. We then specialize this framework to the third level and decompose the cocycle data into symplectic, affine, and phase components, each admitting a cohomological interpretation. As applications, we give a new proof that every two-qudit gate in the third level of the Clifford hierarchy is semi-Clifford, and prove for the first time that every three-qudit gate in the third level is semi-Clifford.

A perfect isometry $I$ (introduced by Broué) between two blocks $b$ and $c$ is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character $\psi$ of $c$ to $\pm$ an irreducible character of $b$. Broué proved that the ratio of the codegrees of $\psi$ and $I(\psi)$ is a rational number with $p$-value zero and that its class in $\mathbb{F}_p$ is independent of $\psi$. This element is called the Broué invariant of $I$ by Boltje. The goal of this paper is to show that if $I$ comes from a Morita equivalence with an endopermutation source $V$, then, up to a sign, the Broué invariant of $I$ is determined by local data of $b$ and $c$. Therefore, up to a sign, it is independent of the endopermutation-source Morita equivalence. Moreover, we show that the sign factor is given by the reduction of the rank of $V$ modulo $p$. As a corollary, we obtain that the Isaacs–Navarro refinement of the Alperin–McKay conjecture holds for inertial blocks.

We show that the numbers of colourings of a link by all finite quandles is equivalent, as an invariant of links, to the profinite completion of the peripheral system. This allows us to reformulate the classical question of the completeness of the former invariant in terms of profinite rigidity. We also develop some profinite quandle theory, and we show that a topological quandle is profinite if and only if it is compact and totally discontinuous and its group of inner automorphism is relatively compact in its homeomorphism group.