Papers

The interval $[1,c]_T$ between the identity element and a Coxeter element $c$ in the absolute order on a Coxeter group $W$ is a generalization of the poset of noncrossing partitions arising when $W$ is the symmetric group. When $W$ is finite, this poset is always a lattice, and it is natural to associate to every element $w\in [1,c]_T$ its moved space $\mathsf{Mov}(w)=\mathrm{Im}(w - \mathrm{Id}_V)$ in the geometric representation $V$ of $W$. It has dimension equal to the reflection length $\ell_T(w)$ of $w$, and gives a realization of $[1,c]_T$ inside the lattice of subspaces of $V$. It is an important tool in the study of $[1,c]_T$. When $W$ is infinite, the moved space of an element $w\in [1,c]_T$ no longer has dimension $\ell_T(w)$ in general, and distinct elements may have the same moved space. We propose a replacement for the moved space of an element $w\in [1,c]_T$ in an arbitrary Coxeter group, that we call absolute moved space of $w$. This subspace $\mathsf{AM}(w)$ of $V$ always contains $\mathsf{Mov}(w)$ and has dimension equal to $\ell_T(w)$, and distinct elements have distinct absolute moved spaces. This allows us to derive several properties of noncrossing partition posets that hold in full generality, and to show that the natural map from $[1,c]_T$ to reflection subgroups of $W$, which to $w\in [1,c]_T$ associates the subgroup $P(w)$ generated by reflections lying below $w$ in the absolute order, is always injective. Among others, we also derive a new proof of the lattice property of $[1,c]_T$ when $W$ has rank three, and exhibit infinitely many new examples of infinite Coxeter groups of rank four and choices of Coxeter elements for which $[1,c]_T$ fails to be a lattice.

We prove that the relative commutant of a finite unitary tuple in a tracial matrix ultraproduct is internal whenever its conjugation operator has a spectral gap off the commutant. This implies the existence of a non-hyperlinear group.

For prosolvable groups, we provide conditions under which two locally conjugate supplements of a normal pronilpotent subgroup are conjugate, analogous to those given by Losey and Stonehewer for finite groups. Our proof relies on a primary-type decomposition of the first cohomology set which may be of independent interest. We also give local conditions for subgroup inclusion and a fixed-point result in the style of Glauberman.

For every integer $n\ge2$, we prove that every countable group embeds in a group of type $\mathrm{FP}_n$. We also construct a group of type $\mathrm{F}_n$ containing a copy of every recursively presented group. Consequently, a finitely generated group embeds in a group of type $\mathrm{F}_n$ if and only if it is recursively presented. This answers questions of Fournier-Facio and Zaremsky, and confirms a suggestion of Gromov. An appendix combines the construction with the controlled-filling argument in a subsequently released OpenAI preprint to prove that every countable group embeds in a two-generator group of type $\mathrm{FP}_\infty$.

We construct a correspondence between ordinary irreducible characters of height zero above the generalized Dade-Glauberman-Nagao correspondence for blocks with central defect. The correspondence is equivariant under group and Galois automorphisms and satisfies a block relation between $\mathcal{H}$-triples. For each pair of corresponding characters, one pair of associated projective representations realizes matching factor sets, equal central scalars, synchronized mixed comparison functions and compatible block correspondences for every intermediate group. The construction uses the multiplicity algebra over the group algebra of the central defect subgroup, a normalized integral lift of a residual semilinear realization, and a graded corner isomorphism. It treats all central-character fibers, including those with nontrivial central character. We also obtain a form suited to normal $p$-sections in an Alperin-McKay-Navarro reduction.

The construction of absolutely irreducible $KG$-modules for a field $K$ and a finite group $G$ is a basic problem in representation theory. For the symmetric group $\operatorname{S}_n$, Gordon James gave an explicit solution using subgroup data associated to a $\mu$-tableau $t$ for partitions $\mu\vdash n$. Here, we want to investigate in which way James's method can be extended beyond symmetric groups. We identify suitable subgroup data from a finite group and use it to construct a module such that an associated factor module is absolutely irreducible or zero.

We develop a logarithmic framework to study the lattice of subgroups of the additive group of rational numbers. By encoding positive rationals via their prime exponent sequences, one obtains a bijection between $\mathbb{Q}^+$ and finitely supported integral sequences indexed by the prime numbers. This correspondence extends to arbitrary subgroups of $(\mathbb{Q},+)$ through a logarithmic greatest common divisor, yielding a classification of subgroups in terms of eventually nonpositive sequences in $\mathbb{Z}\cup \{- \infty\}$. Within this framework, subgroup inclusion, sum, product, and intersection admit simple coordinatewise descriptions, providing a transparent interpretation of the subgroup lattice and recovering several classical results such as the subgroup classification up to isomorphism. Exploiting this perspective, we obtain a complete characterization of the intersection configurations realizable in subgroups of $(\mathbb{Q},+)$. For rank configurations, realizability is characterized by the decreasing condition together with restrictions on the possible ranks and on the minimal zero sets, and, in the finite-support case, by an additional cardinality condition. For binary configurations, this reduces to saying that every decreasing configuration is realizable in the infinite-support case, whereas in the finite-support case the mentioned cardinality condition is required.

Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all completable balanced triangles on any indexed list of at most sixteen nonzero vectors. The second supplies weights for the direction families that can occur in a critical list with no central terms. We give complete coverage arguments, exact certificate files, and separately implemented checking programs. A compression argument removes any need for an induction through smaller list lengths in the sixteen-term potential theorem. The formula for $r\ge4$ remains open.

Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\Q$ containing $\infty$ but not $p$, any continuous odd representation $G_{\Q,S} \to \GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where $G_{\Q,S}$ denotes the Galois group of the maximal extension of $\Q$ unramified outside $S$. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine–Mazur conjecture over $\Q$. Furthermore, we show that for an imaginary quadratic field $K$, any continuous conjugate self-dual two-dimensional $p$-adic representation of its pro-$p$ Galois group $G_{K,S}(p)$ has finite image, provided the primes in $S$ satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's $(3,3)$-conjecture on $p$-class field towers for $p>3$. Namely, we prove that if $K$ is an imaginary quadratic field with $p$-class rank two and the Galois group $G_{K,\varnothing}(p)$ of the maximal unramified $p$-extension of $K$ has Zassenhaus type $(3,3)$, then the $p$-class field tower of $K$ is finite. For $p=3$, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.

Given a free group on $n$ generators, $F_n$, and a topological group $G$, $\operatorname{Aut}\left(F_n\right)$ acts on $\operatorname{Epi}\left(F_n,G\right)$, the set of homomorphisms $f:F_n \rightarrow G$ with dense image. $f\in \operatorname{Epi}\left(F_n,G\right)$ is called redundant if there is a proper free factor $A < F_n$ whose image under $f$ is dense in $G$. For $n \ge 3$, $G = \operatorname{PSL}_2 \left(\mathbb{Q}_p\right)$, we prove that every $f\in \operatorname{Epi}\left(F_n,G\right)$ with torsion-free image is redundant, and give sufficient conditions for $f$ to be redundant even when its image is not torsion-free.

We develop Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\) from the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of \(G\): the orientation-preserving group \(\PGL_2(\ZZ[i])\) and a reflection extension. This distinction explains the two \(G\)-orbits of Farey faces, which merge into one \(\PGL_2\)-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge–face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-$(2+i)$/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.

We introduce endpoint k-path-transitive graphs, in which the automorphism group fixing two prescribed vertices pointwise acts transitively on the paths of length k joining them. We study the geodesic case, where k equals the distance between these vertices. We identify the union of these geodesics with the Hasse graph of a finite bounded graded poset and reduce it to proper blocks by complete cuts andmatching compression. Assume that the induced group K acts faithfully and 2-homogeneously on the first distance layer. We first prove that every nontrivial normal subgroup of K is transitive on every internal layer if and only if soc(K) is. Under this normal-basic condition, we classify the proper blocks in the affine case and the equal-width proper blocks with a 2-transitive internal layer in the almost-simple case. The nontrivial design interfaces are Paley or affine symplectic designs in the affine case, and projective designs, the 2-(11, 5, 2) design, the HigmanSims design, or their complements in the almost-simple case. The proper blocks have reduced rank at most four, whereas almost-simple proper blocks can have arbitrarily large reduced rank without the equal-width condition. We also give a stabilizer factorization criterion for assembling blocks while preserving endpoint-geodesic transitivity.

Reinforcement learning (RL) is a powerful framework for robotic control, yet its practical application is often hindered by high sample complexity. This is particularly restrictive in physical domains where interaction data is costly. While the world often exhibits geometric and physical symmetries, standard RL algorithms typically fail to exploit this structure. In this paper, we demonstrate that exploiting group symmetries significantly reduces the sample complexity of RL. Focusing on finite-horizon Markov decision processes, we find that leveraging homomorphisms induced by group symmetries significantly reduces the theoretical upper and lower bounds on the number of environment interactions required to reach an optimal return. We further extend these bounds to continuous state and action spaces, providing corresponding sample-complexity guarantees under appropriate regularity assumptions. Beyond theory, we validate our findings through controlled experiments and demonstrate the advantages of symmetry-aware policy learning on high-dimensional continuous robotic simulations. Our results show that integrating symmetry into the learning pipeline yields substantial gains in sample efficiency and performance, offering a principled path toward more data-efficient robotics.

Let G be a linear group with Kazhdan property (T) generated by d elements. We show that there are generating sets of size d+2 with arbitrarily small Kazhdan constants. We also prove the same conclusion for every finitely generated group admitting a non-elementary convergence action on a compact metrizable space. This includes non-elementary word-hyperbolic and relatively hyperbolic groups, as well as finitely generated non-elementary subgroups of word-hyperbolic groups.

Let $\Gamma$ be a countable discrete group and let $A$ be a unital $\Gamma$-simple $\Gamma$-$C^*$-algebra. We prove that $A\rtimes_r\Gamma$ has the generalized Powers averaging property if and only if the canonical map $A\rtimes_r\Gamma\to I_\Gamma(A)$ is the unique $\Gamma$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_\Gamma(A)\rtimes_r\Gamma$, with the averaging coefficients still belonging to $A$.

We give a complete classification of finite left-simple skew braces. We prove that a non-trivial finite skew brace has no non-trivial proper left ideals if and only if it is either the trivial skew brace of prime order or the almost trivial skew brace associated with a finite non-abelian simple group.

We extend recent work on analogues of Grün's Lemma from skew-braces to all semiassociative Mal'cev algebras which is a broad class of algebras encompassing all expansions of groups. The direct analogue of Grün's Lemma is known to fail for skew-braces; thus, the more general result is a characterization for perfect algebras of when the upper-central series stabilizes at the second-center. The proof is an application of the description of generating sets for commutators in Mal'cev algebras.

For every group $\Gamma$, we define a homomorphism $d^\Gamma : \mathrm{Comm}(\Gamma) \to \mathbb{Z}^{(\mathcal{FS})}$ from the abstract commensurator $\mathrm{Comm}(\Gamma)$ to the free abelian group $\mathbb{Z}^{(\mathcal{FS})}$ with basis the collection $\mathcal{FS}$ of isomorphism classes of finite simple groups. We investigate the homomorphism $d^\Gamma$ when $\Gamma$ is a finitely generated free group $F$. We explicitly describe the image of $d^F : \mathrm{Comm}(F) \to \mathbb{Z}^{(\mathcal{FS})}$, which is a free abelian group of infinite rank, and we show that the kernel is the monolith of the group $\mathrm{Comm}(F)$. We deduce in particular that every proper quotient of $\mathrm{Comm}(F)$ is abelian. We use this to study the commensurator of a cocompact lattice in the automorphism group of a regular tree, which can be seen as a subgroup of $\mathrm{Comm}(F)$. We show that the image of this group under $d^F$ is again a free abelian group of infinite rank, showing in particular this group is not virtually simple.

Motivated by recent work on conjugacy languages in groups, we introduce a general framework for studying languages of representatives associated to equivalence relations on finitely generated semigroups. After developing this notion in full generality, we particularize it to Green's relations and obtain the corresponding Green languages, which form the central objects of this article. For free inverse monoids, we describe these languages via Munn trees and establish several structural relations among them, including characterizations of the $\mathcal{D}$-, $\mathcal{R}$-, $\mathcal{L}$-, and $\mathcal{H}$-languages. We then investigate the formal language-theoretic complexity of these Green languages. For free inverse monoids of rank at least two, we show that $\mathcal{H}$-Geo is context-free and co-context-free but not regular, while $\mathcal{R}$-Geo, $\mathcal{L}$-Geo, and $\mathcal{D}$-Geo are neither context-free nor co-context-free. In contrast, in the monogenic case, $\mathcal{D}$-Geo is regular and the remaining Green languages are deterministic context-free. We further relate these results to the ShortLex language and the growth of free inverse monoids, proving in particular that $(\text{FIM}_X,X)$ is growth tight. The article concludes with a collection of open problems and directions for future research.

We prove that the weak Haagerup constant is multiplicative with respect to direct products, namely $\Lambda_{\mathrm{WH}}(G \times H)=\Lambda_{\mathrm{WH}}(G) \Lambda_{\mathrm{WH}}(H)$ for any locally compact groups $G$ and $H$. Knudby previously proved the submultiplicative inequality and asked whether equality always holds.

For a finite group $G$, let $\lambda(G)$ denote the minimum number of orbits on the elements of a finite lattice $L$ with $\operatorname{Aut}(L)\cong G$. Babai and Goodman conjectured that $\lambda(G)$ is bounded by an absolute constant. We prove that $\lambda(G)\leq 50$ for every finite group $G$, thereby confirming their conjecture. Moreover, the lattice can be chosen to have a regular orbit. The main algebraic ingredient is a decomposition of a generating set of an arbitrary finite $2$-group into an elementary abelian part and two sets in which no quotient of distinct elements is an involution.

For a finite group $G$, define the field $\mathbb{Q}(G) := \mathbb{Q}(\{\chi(g) : \chi \in \operatorname{Irr}(G) , g \in G\})$. In this article we compute generating sets for the fields $\mathbb{Q}(G)$ when $G$ is a finite general or special linear group, a finite general or special unitary group, or one of the simple groups $\operatorname{PSL}_n(q)$ or $\operatorname{PSU}_n(q)$. We then apply these results to classify all number fields $F$ of degree $2$ or $3$ over $\mathbb{Q}$ such that $F = \mathbb{Q}(S)$ for some non-abelian simple group $S$.

We solve (modulo groups) the word problem for free semigroup satisfying $xy = (xy)^n$. This enables us to show that the variety ${\cal {SCR}}_n$ given by this identity is not equal to the join of the varieties ${\cal{CR}}_n$ and ${\cal{SI}}$ defined by the identities $x = x^n$ and $xy = (xy)^2$ respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice $L({\cal {SCR}}_n)$ is modular does not follow from the earlier results concerning $L({\cal{CR}}_n)$ and $L({\cal{SI}})$. However the variety ${\cal{SO}}_n$ consisting of all semigroups from ${\cal {SCR}}_n$ whose idempotents form a subsemigroup turns out to be equal to the join of the varieties ${\cal O}_n$ of all orthogroups and ${\cal{SI}}$. We deduce from this a description of the lattice $L({\cal{SO}}_n)$ modulo groups.

We prove that every acylindrical action on uniformly locally finite product of trees induces the hyperfinite orbit equivalence relation on the Roller boundary. As a byproduct, we construct an example of a standard Borel space and two commuting bounded-to-one surjective partial Borel maps that generate a universal countable Borel equivalence relation. This contrasts to Shinko-Weilacher-Yu's theorem on hyperfiniteness of bounded-to-one actions of commutative monoids.

We prove that if $G$ is a residually finite group, $\widehat{G}$ is its profinite completion and the map $H_2(G, \mathbb{Z}) $ $ \to H_2(\widehat{G}, \widehat{\mathbb{Z}}),$ induced by the canonical map $G \to \widehat{G}$, is injective, then the non-abelian exterior square $G \wedge G$ is residually finite. We show that if $G$ is a finitely presented centre-by-metabelian group then $\nu(G)$, the non-abelian tensor product $G \otimes G$ and the non-abelian tensor square $G \wedge G$ are residually finite. Furthermore we prove that if $G$ be a finitely presented metabelian group then the weak commutative construction $\X(G)$ is residually finite. We discuss a criterion for the $q$-exterior square $G \wedge^q G$ to be residually finite.

We construct the Conway-Wales lattice from a self-dual even code $C$ over $\mathbb{Z}_4$ whose binary layers interweave the Hamming codes $\mathcal{H}_7$ and $\mathcal{H}_8$. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order $583\,704\,576\,000$. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order $17\,971\,200$. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem.

Let $G=\langle a,b\rangle$ be a finite group with distinct nonidentity generators, and suppose that $H=\langle ba^{-1}\rangle$ is normal. We determine exactly which complementary generator choices on the nonterminal cosets of $H$ extend to two arc-disjoint Hamiltonian paths with both terminal vertices in $H$. The associated return residue must avoid an explicit set of at most six residues. When $[G:H]\ge3$, any two nonterminal cosets may be left free while all other choices are prescribed; two is best possible. Every such Cayley digraph has two arc-disjoint Hamiltonian paths; this includes all two-generator Cayley digraphs of finite abelian groups. The criterion follows from a complete distance formula for the Hamiltonian cut spectra of $\mathrm{Cay}(\mathbb{Z}_q;\{D,D+1\})$. Their unique nondegenerate exception comes from the odd central coefficient $(q-1)/2$. The geometric ingredient is a confinement theorem: two sector-filled integer sequences satisfying a length inequality, with overlapping ranges and distance at least two, have one sequence contained in a single gap of the other. Sector filling bounds alternation, and the length inequality excludes interlacing. A self-contained proof of the primitive-ray cycle profile follows the intersection geometry of Curran and Witte.

In the Kourovka Notebook, V.\,D.~Mazurov asked whether an automorphism $\alpha$ of prime order $m$ of the infinite free Burnside group $G=B(m,q)$ of prime period $q$ that cyclically permutes the free generators of $G$ must have a nontrivial fixed point. In this paper, we show that for $m\ge3$ the answer is negative for all sufficiently large prime periods $q$. Precisely, we prove that for any odd $m\ge3$, there exists $q_m>0$ such that for every prime period $q>q_m$, the automorphism of $B(m,q)$ of order $m$ that cyclically permutes its free generators is fixed-point-free.

Every group $G$, of arbitrary cardinality, admits a right-invariant metric with at most five values whose isometry group is exactly the group $\Xi(G)$ of permutations preserving every inverse-pair difference. Five is sharp. Writing $N(G)$ for the least number of values, including zero, we obtain $N(G)=1$ for the trivial group, $N(G)=2$ for $C_2$ and $C_3$, $N(G)=3$ for every other group outside an explicitly listed set $\mathcal{E}$ of 24 finite groups, $N(G)=4$ on $\mathcal{E}\setminus\{C_2^3, C_3^2\}$, and $N(C_2^3)=N(C_3^2)=5$. The exact result combines Cayley-graph realization theorems with verified finite certificates and hand proofs of the lower bounds. Separately, a uniform constructive 32-value bound uses subgroup-entry ranks, sign-variation conflict labels, and short-word rigidity, but no Cayley-index input or computation. In Knill's right-translation sense, a right-invariant metric on $G$ is naturalizing when every group law on its underlying set with isometric right translations is isomorphic to $G$; a group is natural when it admits such a metric. We prove that an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. Every natural group admits a naturalizing metric with at most five values, and this bound is sharp.

In this thesis, we study finitely generated groups close to wreath products, all defined as semi-direct products, from various perspectives. The first main result is a quasi-isometric classification of some permutational wreath products. The strategy we follow relies on recent work of Genevois and Tessera, but yields a stronger rigidity phenomenon: under suitable assumptions, a quasi-isometry between two permutational lamplighters always induces a quasi-isometry of pairs between the base groups equipped with the corresponding normal subgroups. We also show that, in some specific cases, being quasi-isometric is equivalent to being bijectively quasi-isometric. A chapter is also dedicated to the study of standard wreath products whose lamp groups have polynomial growth. We establish a rigidity result for quasi-isometries between such wreath products: they must all be {\guillemotleft}measure-scaling{\guillemotright}. In particular, self-quasi-isometries of such wreath products all lie at bounded distance from bijections. We also prove that this class of lamplighters provides examples of amenable quasi-isometric groups that are not bijectively quasi-isometric. The second part of this thesis relies on joint works with Corentin Correia, in which we study halo products of Genevois and Tessera from a measured point of view. We construct orbit equivalence couplings between two halo products of the same kind and, using the recent technology of Folner tilings, we exhibit couplings between halo products and free abelian groups. We also establish several asymptotic estimates of the isoperimetric profiles of these groups, improving previous results of Erschler-Zheng and Saloff-Coste-Zheng. These computations allow us to prove that, in many cases, the orbit equivalence couplings thus obtained are quantitatively optimal.