Papers

We show that for any recognizing $S$-machine $\textbf{S}$ with superadditive time function $f$, there exists a finitely presented group whose conjugator length function is quadratic and whose Dehn function grows like the square of $f$. This result is dual to that of a previous paper of the authors, which constructed a family of groups with cubic Dehn function that have various conjugator length functions. We thereby give the first known example of a finitely presented group whose conjugator length function is recursive, but whose Word and Conjugacy Problems are both undecidable. This answers an analogue of a question of Rips. Moreover, given a finitely presented group $G$ with decidable Word Problem, we obtain a finitely presented group with decidable Conjugacy Problem whose Dehn function grows faster than that of $G$. Finally, combining this result with its dual, for a wide array of pairs of functions $(f,g)$ we furnish an example of a finitely presented group with Dehn function equivalent to $f$ and conjugator length function equivalent to $g$. This shows that the two invariants are very strongly independent and making significant progress on a question of Bridson, Riley, and Sale.

We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.

Let $G$ be a permutation group that acts $2$-transitively on the finite set $V$ and let $\mathcal{T}=(V,T)$ be a two-graph whose automorphism group contains $G$. In this paper, we classify those strongly regular graphs $\Gamma$ with vertex set $V$ whose automorphism group is a transitive maximal subgroup of $G$ and whose associated two-graph is $\mathcal{T}$. In doing so, we obtain a new family of vertex-transitive strongly regular graphs whose associated two-graph arises from $P\Sigma L(2,q)$.

A classification is given of regular, rotary, and birotary Cayley maps of which the vertex number is coprime to the valency. The classification is involved in constructions of new examples of interesting Cayley maps.

Given a discrete group $\Gamma$ acting on a compact space $X$, and a stabilizer subgroup $\Lambda\leq \Gamma$ of $X$, we use the Rieffel induction of covariant $(\Lambda, X)$-representations to study classes of representations induced by certain characters on $\Lambda$ and the $\text{C}^*$-algebras they generate. When $X$ is a $\Gamma$-boundary, we obtain new classes of simple, traceless group $\text{C}^*$-algebras. When the $\Gamma$-boundary $X$ is an extreme boundary, we show that the associated group $\text{C}^*$-algebras are also purely infinite, answering part of a question posed by Kalantar and Scarparo on $\text{C}^*$-algebras associated with Thompson's groups. The latter $\text{C}^*$-algebras are shown to be selfless in the sense of Robert.

Let $\theta$ be an ordinal and let $(N_\alpha)_{\alpha\leq\theta}$ be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group $G$, with $N_0=\{e\}$. Write $\rho G$ for the Raikov completion of $G$, and put $K_\alpha=\overline{N_\alpha}^{\rho G}$ and $H_\alpha=GK_\alpha$. The successive and limit differences of the groups $H_\alpha$ mark the first stages at which points of $\rho G$ become old quotient points. For every nonzero ordinal $\theta$ and every set $S\subseteq(0,\theta]$, there is a Hausdorff pseudocompact Boolean group with a continuous tower of closed normal subgroups such that every member of the tower is pseudocompact and the nonempty strata occur exactly at the stages in $S$. The terminal quotient can simultaneously be chosen isomorphic to $\mathbb Z/2\mathbb Z$, and for every $\delta<\theta$ the successive factor $N_{\delta+1}/N_\delta$ is compact exactly when $\delta+1\notin S$. Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. For a closed normal subgroup $N$ of $G$, let $\widehat q:\rho G\to\rho(G/N)$ extend the quotient homomorphism and put $K=\overline N^{\rho G}$. Over $gN\in G/N$, the fiber in $\rho G\setminus G$ is $g(K\setminus N)$, homeomorphic to $\rho N\setminus N$, while for $y\in\widehat q(\rho G)\setminus(G/N)$ the entire $\widehat q$-fiber lies in $\rho G\setminus G$ and is homeomorphic to $\rho N$. For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. Further results concern pseudocompact Raikov remainders and include countable Abelian examples.

Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for $S_n$; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$. The correct geometric invariant of a point $x\in K^n$ is a pair $(\mu,z)$: a classical partition $\mu$ on the $r$-th powers of the nonzero coordinates, together with the number $z$ of zero coordinates. We construct a $G(r,1,n)$-invariant ideal $I_{(\mu,z)}$ for each such pair, and show that a dominance order on $(\mu,z)$, with $z$ taking strict priority over $\mu$, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical $B_n$-Specht ideal theory at $r=2$ when $z=0$, and establish radicality of $I_{(\mu,z)}$ at the two extremes of the $z=0$ level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.

Clark's syntactic concept lattice (SCL) records two-sided distributional structure, and Wurm extended it to tuples of arbitrary finite arity. We study \(\operatorname{cmp}_f(L)\), the minimum image size of a finite-monoid observation that preserves guarded tuple substitution through arity \(f\) on the principal layer. For regular languages, we characterize \(\operatorname{cmp}_f(L)\) exactly as the least cardinality of the codomain of an \(f\)-separating relational morphism from the pointed syntactic monoid. Let \(\operatorname{ch}(\mathbf V)\) denote the least arity at which these compression numbers stabilize uniformly over a pseudovariety \(\mathbf V\). Our main result is the following trichotomy of possible uniform heights: \(\operatorname{ch}(\mathbf V)\in\{1,2,\infty\}\), with \(\operatorname{ch}(\mathbf V)=\infty\) if and only if \(\operatorname{Synt}(\{ab\})\in\mathbf V\). Thus no finite uniform compression height \(3,4,\ldots\) occurs. The infinite case is sharp: inside \(\langle\operatorname{Synt}(\{ab\})\rangle\), every boundary \(d\to d+1\) admits unbounded compression gaps, and arbitrary finite strict prefixes of the arity hierarchy are realizable. On the finite side, commutative monoids and bands stabilize at arity one, while every completely regular syntactic monoid stabilizes by arity two; finite group kernels show that the binary bound is sharp. At unary arity, every nonempty finite simple graph is realized by an explicit length-three language, yielding an exact chromatic-number formula and NP-completeness of deciding \(\operatorname{cmp}_1(L)\le 3\) for explicitly listed length-three languages. The structural boundary between compression heights one and two remains open.

It was conjectured that the nonzero idempotents of the integral quandle ring of a finite latin quandle are trivial. We prove this holds for latin dihedral quandles. Furthermore, we discuss counterexamples against this conjecture.

The orbit problem is the problem of whether, given $x, y\in \mathbb{Q}^n$ and an $n\times n$ matrix $A$, there exists an $i\in\mathbb{N}$ such that $A^ix = y$. In 1980, Kannan and Lipton proved that the orbit problem is decidable. We show that a generalization of the orbit problem, where the field is $\mathbb{Q}(X)$ for $X$ a countable set of transcendentals, is also decidable. We define the orbit power problem for an arbitrary group $G$ to be the problem of when, given $x, y\in G$, there exists an $n\in\mathbb{Z}$ such that $x^n = y$. We then use the main result to show that the power orbit problem is decidable for an assortment of groups, including the braid groups and $\operatorname{Aut}(F_2)$.

We give solutions to several problems in combinatorial group theory recorded as open in the literature. Our main source is the Kourovka Notebook, although we also consider questions from elsewhere. All solutions were found with the assistance of large language models and checked by the authors.

The affine Conway group is the group of affine isometries of the Leech lattice. This group is isomorphic to the automorphism group of a standard fundamental domain for the action of the Weyl group on the even hyperbolic lattice II_{1, 25} of rank 26. In this paper, we show that the affine Conway group has exactly nine conjugacy classes of involutions. We investigate their properties and show that, among these nine classes, one class can be regarded as an analogue of the class of Enriques involutions of K3 surfaces. Motivated by possible applications to K3 and Enriques surfaces, we investigate in detail the orthogonal groups of the hyperbolic lattices arising as the invariant sublattices of involutions in II_{1, 25}. This computation is carried out using the Borcherds method. Unlike the examples considered previously, the induced chambers possess infinitely many walls.

Let $\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let $W=\left(\bigoplus_{G/\Gamma} \mathbb{Z}/2\mathbb{Z}\right) \rtimes G.$ We show that $W$ is not an MF group and that $C^*(G)$ is not a finite C*-algebra. The ideas and proofs were generated by ChatGPT 5.6 Sol, we have only refined their arguments in a hopefully more palatable form.

We prove Mühlherr's Twist Conjecture: any two angle-compatible Coxeter generating sets of a Coxeter group differ by a finite sequence of elementary twists and a conjugation. Combined with earlier work of Howlett-Mühlherr and Marquis-Mühlherr, this completes the resolution of the Isomorphism Problem for Coxeter groups. A further consequence is that ${\rm Aut}(W)$ is finitely generated for every Coxeter group $W$, and there is an algorithm producing a finite set of generators for ${\rm Aut}(W)$ starting from any Coxeter matrix. Of the vast literature on the Twist Conjecture, we utilise only two results in an essential way: strong rigidity of $2$–spherical Coxeter systems, due to Caprace and Mühlherr, and the framework of markings and hierarchies developed by Caprace and Przytycki for the twist-rigid case. We also exploit in a fundamental way some soft ideas from JSJ theory and an observation of Mihalik-Tschantz on splittings of Coxeter groups. No form of AI was used in the writing of this manuscript, nor in the research that it presents.

Bonk and Kleiner proved that if $G$ is a Gromov hyperbolic group whose boundary $\partial_{\infty}G$ is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, and the Ahlfors regular conformal dimension of $Z$ is attained and equal to $Q$, then $G$ acts discretely, cocompactly, and isometrically on $\mathbb{H}^3$. In this article, we extend the Bonk-Kleiner theorem to the setting of relatively hyperbolic groups. More precisely, we prove that if $(G,\mathcal{H})$ is a relatively hyperbolic group whose Bowditch boundary is homeomorphic to an Ahlfors $Q$-regular metric $2$-sphere $Z$, with the Ahlfors regular conformal dimension of $Z$ attained and equal to $Q$, then $G$ acts discretely and isometrically on $\mathbb{H}^3$, and every subgroup in $\mathcal{H}$ is virtually $\mathbb Z^2$.

The topological Halmos–von Neumann theorem is a fundamental result of topological dynamics that completely classifies continuous actions of abelian topological groups which are ergodic and have discrete spectrum. It rests on a correspondence between such actions (up to the choice of a distinguished point), compactifications of the acting group and subgroups of its dual group. We generalize both the classical statement and the underlying categorical equivalences to the framework of open group bundles over locally compact base spaces. As a byproduct, we establish a Pontryagin-type duality between étale Hausdorff abelian group bundles and open proper Hausdorff abelian group bundles, extending the classical duality between discrete and compact abelian groups.

Theory of Rota–Baxter operators on rings and algebras has been developed since 1960. In 2021, L. Guo, H. Lang, Y. Sheng have defined the notion of Rota—Baxter operator on a group. We classify all Rota—Baxter operators on the finite Heisenberg group and the extraspecial group of order $p^3$. We also investigate an analogue of the spectral property for Rota—Baxter operators on algebras.

We show that the homeomorphism groups of the following spaces have precisely $2^{2^{\aleph_0}}$ maximal subgroups: the rational numbers $\mathbb{Q}$, the Baire space $\mathbb{N}^{\mathbb{N}}$, the space $\mathbb{N}\times 2^{\mathbb{N}}$ where $2^{\mathbb{N}}$ is the Cantor set, the ordinal $\omega^2$ under its order topology, and the Sorgenfrey line $\mathbb{S}$. More generally, we find sufficient conditions on a group $G$ acting on a topological space which imply that $G$ has at least $2^{2^{\aleph_0}}$ maximal subgroups. Moreover, if the groups $\operatorname{Homeo}(\mathbb{Q})$ and $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ are equipped with the pointwise topology, then it is shown that $\operatorname{Homeo}(\mathbb{N}^\mathbb{N})$ has precisely $2^{\aleph_0}$ open maximal subgroups, and $\operatorname{Homeo}(\mathbb{Q})$ has precisely $\aleph_0$ open maximal subgroups and $2^{\aleph_0}$ closed maximal subgroups.

Our results implies, in particular, that a finitely generated solvable group $G$ is nilpotent if and only if it contains a solution to any unimodular equation, i.e., an equation of the form $\prod g_ix^{n_i}=1$, where $g_i\in G$ and $\sum n_i=\pm1$. A similar fact turns out to be true for Lie algebras. We also exhibit an example of a unimodular equation $w(x)=g$ over a finitely generated group $G$, which has a solution (in $G$) for any $g\in G$, but the solution is not unique for some $g\in G$. We show that, for nilpotent groups $G$, the set of unimodular mappings $G^n\to G^n$ (which are defined naturally) forms a group under the composition.

Recently, two related papers [arXiv:2412.13941, arXiv:2409.03626] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [arXiv:1410.2540] using stackings and by [arXiv:1410.2579] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [arXiv:2412.13941] and [arXiv:2409.03626]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two.

A classical theorem of Gaschütz states that an abelian normal subgroup N of a finite group G has a complement in G whenever it has a complement in some subgroup H with $N \leq H \leq G$ and gcd(|N|, [G:H]) = 1. We characterize the finite groups N for which this theorem remains valid without the abelian hypothesis: this is the case if and only if $Z(N) \cap N' = 1$ and N has a complement in a quotient of the holomorph Hol(N) by a diagonal copy of N'. This completes a previous paper by the second author.

Let $\pi$ be a set of primes and let $G$ be a finite $\pi$-separable group. We prove that the Cartan matrix $C$ for the $\pi$-partial characters of $G$ is equivalent over the integers to a matrix $\operatorname{diag}(|\mathbf{C}_G(x)|_{\pi'})$, where $x$ runs over a set of representatives of the $\pi$-classes of $G$. In particular, we prove that $\det C = \prod |\mathbf{C}_G(x)|_{\pi'}.$

We show that power semigroups of groups, and more generally reduced finitary power monoids, arise naturally as convolution monoids in Kleisli categories of powerset monads: (1) for the non-empty powerset monad, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_+)}(1,G)$ is isomorphic to the power monoid $\mathcal P_+(G)$; (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}$ $(\mathbb Z/2\mathbb Z,H)$ is isomorphic to the reduced finitary power monoid $\mathcal P_{\mathrm{fin},1}(H)$. This unifies several constructions in power semigroup theory: Kleisli convolution representations of semigroups, base change along surjective group homomorphisms, and rigidity of automorphism groups. As an application, we prove that for every proper numerical monoid $S$, the Kleisli Hom-monoid $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}(\mathbb Z/2\mathbb Z,S)$ is rigid. This gives a proof of the Tringali–Yan conjecture in the language of Kleisli categories. It should be mentioned that this conjecture was already proved by Bhowmik and Tringali in a preprint posted on arXiv on July 25, 2026.

We show that if $G$ is a real semisimple Lie group and $\Gamma<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $\rho$, then $\Gamma$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-covolume discrete subgroups that act ergodically on the Furstenberg boundary of $G$, providing a counterexample to a conjecture of Margulis for $G = SO(n,2), n > 2$. The examples arise from lattices $\Gamma<H= SO(n,1)$ and their deformations in $G$. Perhaps more importantly, we describe a new approach to studying deformations of such lattices by relating them to deformations of the smooth right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$. This correspondence allows us to apply recent results of DeWitt and Dolgopyat on smooth group actions and to give examples where the right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$ can fail to be $C^0$-locally rigid.

We prove that mapping class groups of surfaces and of locally finite connected graphs support a unique Polish group structure.

The Andreadakis-Johnson filtration and its associated construction, known as the Johnson homomorphism, are useful tools in group theory. They provide a step-by-step approach to studying the automorphisms of a given group. After explaining the basics we survey both classical and recent results around the Andreadakis-Johnson filtration, with emphasis on the mapping class group of a once-bordered surface.

Let $W$ be an affine Coxeter group of type $\widetilde{A}_n$, that is, the affine symmetric group $\widetilde{S}_N$ with $N=n+1$, let $T$ be its set of reflections, and let $\mathrm{Red}_T(w)$ be the set of reduced reflection factorizations of an element $w\in W$. The braid group acts on $\mathrm{Red}_T(w)$ by the Hurwitz action. For finite Coxeter groups it is known exactly when this action is transitive, namely precisely for the parabolic quasi-Coxeter elements. We address this problem for the affine type $\widetilde{A}_n$ by determining all orbits of $\mathrm{Red}_T(w)$.

We introduce a centroid-based block decomposition of the left-regular representation and use it to prove operator-valued Haagerup inequalities for finitely generated groups with centroid maps. The decomposition separates the three centroid growth conditions into distinct operator-norm contributions and, under a factorization hypothesis for the associated Schur multipliers, yields complementary lower bounds. For median groups of rank $\nu$, the centroid blocks are related to spherical operators, giving two-sided estimates with the explicit factor $\binom{\nu+r}{r}$; this applies in particular to right-angled Artin groups and to groups acting on finite-rank $\operatorname{CAT}(0)$ cube complexes. For finite-rank coarse median groups, we construct a coarse iterate median with error independent of the cardinality of the input set and obtain spherical-block estimates with factor $o(r^{1+\nu/2+\varepsilon})$ for every $\varepsilon>0$.

We study real ideal points of $SL_2(\mathbb C)$-character varieties of knot exteriors containing essential Conway spheres. Under explicit deformation-theoretic hypotheses on the two complementary tangles, we show that essential Conway spheres are detected via Culler-Shalen theory with real ideal points; such ideal points are then deformed into arcs of $SL_2(\mathbb R)$ representations on both sides. We then compute the asymptotic behavior of the representations on these arcs and compute the translation numbers of their lifts to $\widetilde{PSL}_2(\mathbb R)$ representations. Combining this with a result of Gao, we conclude that for certain knots with essential Conway spheres, all sufficiently large positive and negative rational fillings have left-orderable fundamental group. This verifies the $L$-space conjecture for large-slope Dehn fillings of knots which were previously unknown in the literature. We verify the criteria for three infinite families of knots assembled from torus-trivial and twist-trivial tangles, then count the resulting real ideal points, determine the longitudinal translation numbers of all constructed branches, and identify exactly the zero-translation arcs. Computations for the $28$ distinct verified census exteriors listed in Table \ref{tab:instances} agree with the certified branches.

This PhD thesis studies the classification of group extensions, whose main result is the classification, up to isomorphism, of finite nonsolvable groups of order less than 23,040. To make this classification computationally effective, we develop reduction methods that replace extension computations involving a finite nonsolvable group by computations in suitable smaller solvable subgroups. This reduction from nonsolvable to solvable computations led to a major decrease in computing time and made it possible, on a standard personal computer available in 2002, to classify around 8.4 million nonsolvable groups of order less than 23,040. A nonsolvable group E is constructed from a given perfect kernel P, its perfect residual, extended by a given solvable quotient H=E/P. The isomorphism problem for the resulting nonsolvable groups E is reduced to isomorphism computations between extensions involving a small nilpotent subgroup U<P and the quotient H. We introduce and use (Z\phi)-extensions and supplement methods to obtain explicit classifications in a range of finite cases. These methods are applied to the classification of finite nonsolvable groups of order less than 23,040 and provide algorithms for constructing and identifying extensions up to isomorphism.