Papers

We show that the existence of a growth gap for infinite-index subgroups of a given finitely genrated group can depend on the finite generating set. More precisely, for any irreducible lattice $\Lambda$ in a higher rank semisimple Lie group $G$ with Kazhdan's property (T), the group $\Lambda \times \Lambda$ admits one finite symmetric generating set with a growth gap and another without a growth gap. We also prove that the growth gap can be made arbitrarily small. In contrast, for a non-elementary hyperbolic group the existence of a growth gap is independent of the finite generating set.

The goal of this article is to present a computation of the Galois cohomology of the group $G = \mathrm{PGL}_2$ over rings of Laurent polynomials. This computation recovers the main result of \cite{CGP} in this case by a new method, based on the analysis of actions on appropriate geometric objects (products of trees and, in general, of affine buildings), which was already used in \cite{ARR} to give a new proof of the theorem of Raghunathan-Ramanathan \cite{RR} concerning Galois cohomology over polynomial rings. Going beyond Galois cohomology, this method also enables one to determine the finite subgroups in the group of points over relevant polynomial rings. In view of these and other potential applications of the method in different situations (in particular, in the study of algebraic groups over the coordinate rings of general affine curves), we have attempted to make our exposition largely self-contained and accessible to broad mathematical audience.

We investigate groups $\hat{G}$ given by a presentation $\mathcal{P}=\langle \textbf{x}: \textbf{r} \rangle$ whose relators $\textbf{r} \subseteq F(\textbf{x})$ are comprised of a set of subwords in $F(\textbf{x})$, i.e. $\textbf{r}$ admits a $\textbf{u}$-substitution in the sense that there exists a homomorphsim $\epsilon: F(\textbf{u}) \rightarrow F(\textbf{x})$ and a subset $\mathbf{v} \subseteq F(\textbf{u})$ such that $\textbf{r}=\epsilon(\textbf{v})$. Equivalently, $\mathcal{P}= \langle \textbf{x}: \epsilon (\textbf{v}) \rangle$ is referred to as the composition of the presentation $\mathcal{G}= \langle \textbf{u}: \textbf{v} \rangle$ with $\mathcal{H}= \langle \textbf{x}: \epsilon(\textbf{u}) \rangle$ of the groups $G$ and $H$, respectively. We survey known results and record structural properties which do not explicitly appear in the literature, e.g. that there is the relative presentation $\langle G, \textbf{x}: \textbf{u}= \epsilon(\textbf{u}) \rangle$ for $\hat{G}$. Thus there is a natural map $\epsilon: G \rightarrow \hat{G}$ and we may consider the associated problems (e.g. injectivity, finiteness). As an application, we investigate the class of groups $\mathcal{G}(\mathcal{B})$ obtained from substituting a presentation of the trivial group into a deficiency one group presentation for $\mathbb{Z}$. Our results show $\mathcal{G}(\mathcal{B})$ is a proper subset of the class of 2-knot groups and properly contains all classical knot groups. A subfamily is investigated which includes the group of the trefoil knot and uses Higman's presentations for the trivial group.

We show that for a discrete isometry subgroup acting on a proper CAT(-1) space X, if the critical exponent is less than $1$, then the critical exponent equals the Hausdorff dimension of the entire limit set. Consequently, the limit set must be a Cantor set. As an application, we prove that any finitely generated, torsion-free discrete subgroup in Isom(X) with critical exponent less than one must be geometrically finite and free. This answers a question of Kapovich.

Generalising Hölder's classical group enumeration for squarefree orders (1895), we provide an exact formula for the number of isomorphism types of groups of a given cubefree order. After more than 130 years, this is the first such formula that covers significantly more orders than the squarefree ones (83% versus 61% of all integers). Like Hölder's formula, ours is combinatorial: it can be evaluated from the prime factorisation of the order by arithmetic operations and table look-ups, without constructing a single group. The structure of our formula leads to counting formulas for natural subclasses of cubefree groups, with applications in computational group theory. We also derive new asymptotic results. Blackburn et al. (2007) conjectured that the number gnu(n) of groups of cubefree order n satisfies gnu(n)<n^2. We show that gnu(n)\leq n^{2+o(1)}, which improves the bound gnu(n)<n^8 recorded in their survey, and we prove that the exponent 2 is best possible, that is, gnu(n)\geq n^{2-o(1)} for infinitely many cubefree n. Lastly, we show that a much stronger form of the conjecture holds for almost every cubefree order, namely, \gnu(n)\leq (\log n)^{(\log\log n)^{O(1)}}.

For a finite group $G$, the representation dimension $\delta(G)$ is the least dimension of a faithful complex representation of $G$. We prove that $\delta(H\times K) = \delta(H) + \delta(K)$ whenever $H$ and $K$ are $p$-groups for a fixed prime $p$, and show by example that this additivity can fail more generally for nilpotent groups. We also determine $\delta(G)$ for several important classes of non-abelian $p$-groups, viz. VZ groups, Camina groups, and metacyclic groups; and compute $\delta(G)$ for all groups of order $p^6$ ($p\geq 5$), organized by their isoclinism family.

By a short adaptation of an argument of Singerman, we show that for every prime power $q$, every orientably regular map (or hypermap) whose orientation-preserving automorphism group is isomorphic to PGL(2,q) is reflexible.

A finite group has the weak Erdos-Ko-Rado property if all of its transitive permutation actions have the EKR property. We characterize this property in terms of normal subgroups and chief factors. More precisely, we introduce a local intersection density and establish a normal-extension criterion which reduces the weak EKR property to difference-set conditions on the elementary abelian chief factors and the linear groups induced on them. For chief factors of rank one the condition is automatically satisfied, and for chief factors of rank two, this condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces. In the second part of the paper, we solve an open problem by determining the possible intersection densities of transitive permutation groups with a prescribed point stabilizer. We prove that, for every finite group $H$ of order $m\geq 4$ and every integer $n\geq m$, there exists a faithful transitive permutation group with point stabilizer isomorphic to $H$ and intersection density $n/m$.

For any finite group, we pose the following question: given an irreducible real representation, is the associated representation sphere non-orientable if and only if the representation is nontrivial of real type? We prove that the answer to this question is yes if the group has a normal Sylow 2-subgroup, but exhibit a 2-nilpotent group of order 112 for which the answer is no via elementary arguments. We also link the question to the unit group of the Burnside ring, where we recover a basis discovered by Bouc for 2-groups, and pose a conjecture about detection of non-orientability from solvable subgroups. The question is motivated by issues arising from the construction of permutation twisted cohomology for finite groups.

We study the critical exponent of the Falconer functional for projective Anosov representations with Lipschitz limit sets and its relationship to the Hausdorff dimension of the limit set. Extending a result of Pozzetti, Sambarino, and Wienhard, we show, under density and irreducibility assumptions on the representation, that this exponent equals the Hausdorff dimension of the limit set. We also show that these assumptions are necessary for the strategy used there and in the present paper. To this end, we study the action of $\mathrm{SO}(p,p)$ on the space of maximal isotropic subspaces of $\mathbb R^{p,p}$ and provide examples of $\mathbf H^{p,q}$-convex-cocompact representations of lattices with virtual cohomological dimension $p$. Besides yielding counterexamples to seemingly innocent claims about irreducible representations in $\mathrm{SO}(p,q+1)$, these examples are of independent interest: they admit equivariant spacelike embeddings of Riemannian symmetric spaces into $\mathbf H^{p,q}$ on which uniform lattices act cocompactly.

We prove that every finite simple group of Lie type over a field of order at most 8 contains a conjugacy class whose square is the whole group. Together with the previously established cases, our results complete the proof of Thompson's conjecture. We explicitly construct the required class, address all low-rank identifications, and provide the precise computational certificates used in the exceptional character calculations.

A finite group is totally $3$-closed if every faithful permutation representation of it is $3$-closed. We study this property for the finite simple projective special linear groups. We prove that $\PSL_2(q)$ is totally $3$-closed if and only if $q\geq 7$ is prime, and that $\PSL_3(q)$ is totally $3$-closed if and only if either $q=3$, or $q$ is prime and $q\equiv 2\pmod 3$. We further prove that $\PSL_4(q)$ is never totally $3$-closed and that $\PSL_n(q)$ is not totally $3$-closed whenever $n\geq 5$ and $q>2$. Within the family $\PSL_n(q)$, only the groups $\PSL_n(2)$ with $n\geq 5$ remain unresolved. In particular, this answers Problem~20.2 of the Kourovka Notebook affirmatively.

Among finite simple groups, character values of alternating and sporadic groups have relatively low irrationality at any prime $p$, whereas those of simple groups of Lie type can have arbitrarily high $p$-irrationality. We provide concrete evidence supporting this phenomenon. In particular, we show that if $\chi:=R_{\mathbf{T}}^{\mathbf{G}}(\theta)$ is a Deligne–Lusztig character of a finite reductive group $\mathbf{G}^F$, with $\theta$ an irreducible character of a maximal torus $\mathbf{T}^F$, and if $\chi$ has degree prime to $p$, then the so-called $p$-rationality level of $\chi$ coincides precisely with that of $\theta$. We present further evidence suggesting that Lusztig induction preserves $p$-rationality for characters of $p'$-degree.

For every finite-rank crystallographic affine Coxeter system $(W,S)$ and Coxeter element $c$, we construct a minimal linear graded free resolution of the trivial module over $k[M([1,c]_T)]$ supported on a rectified exceptional cluster complex. Hence the affine dual braid monoid algebra is Koszul over every field $k$. The exceptional complex is introduced to recover the principal-fibre topology missing from the direct Reading–Stella labelling. Half-orbit rectification replaces the transjective root labels by ordinary exceptional modules, so that a face $F$ determines an exceptional wide subcategory and the intrinsic weight \[ \omega(F)=\operatorname{cox}(\operatorname{wide}\langle F\rangle). \] The resulting principal fibres are induced subcomplexes and split canonically as joins of subcomplexes attached to connected Dynkin and affine blocks; these subcomplexes are contractible. Affine non-lattice divisibility creates the genuinely nonprincipal case. The McCammond–Sulway completion shows that whenever no greatest interval right divisor exists, all maximal interval right divisors share a common nontrivial complete finite Coxeter component. In the associated exceptional-wide decompositions, this common Coxeter component is the Coxeter element of a Dynkin block, and the subcomplex attached to that block occurs as a common contractible join factor. Thus every nonidentity fibre is contractible, and the weighted-face complex is exact, minimal and linear. In particular, $\operatorname{Tor}^{A_c}_q(k,k)$ is indexed by $q$-vertex exceptional cluster faces in internal degree $q$, and $\operatorname{pd}_{A_c}k=|S|$.

We show that the $ \mathrm{K}_1 $-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass–Serre dimension and, for the Tits index $ \mathsf{E}_{8, 2}^{78} $, contains a field. For classical or globally isotropic reductive groups schemes $ \mathrm{K}_1 $ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

It was known at least to L.E. Dickson in 1901 that $\mathrm{SL}(2,q)$, in its natural action on $\mathbb{F}_q^2\setminus\{0\}$, has a sharply transitive subgroup only when $q\in\{2,3,5,7,11\}$. For $q$ prime, this result stems from Galois' letter to Chevalier in 1832. We extend this result to sharply transitive subsets of $\mathrm{SL}(2,q)$ and show that they only exist when $q\in\{2,3,5,7,11\}$.

We prove in ZFC that every torsion-free Abelian group of cardinality $\mathfrak c$ admits a Hausdorff countably compact group topology without nontrivial convergent sequences. In particular, this applies to the free Abelian group $\mathbb{Z}^{(\mathfrak c)}$, the Baer-Specker group $\mathbb{Z}^\omega$ and $\mathbb{Q}^{(\mathfrak c)}$. For the topology constructed on $\mathbb{Z}^{(\mathfrak c)}$, the coordinatewise nonnegative cone is countably compact in the subspace topology. Consequently, there exists in ZFC a commutative Tychonoff countably compact topological semigroup which has two-sided cancellation but is not a group, giving a negative answer to Wallace's question. Combined with earlier results, the main theorem also yields in ZFC a Tychonoff countably compact topological semigroup containing a copy of the bicyclic semigroup and a functionally Hausdorff countably compact paratopological group that is not a topological group.

Let $F_m=F(a_1,\dots,a_m)$ with $m\ge 2$, and fix $q\ge 1$. For every fixed $0<\lambda<1$, we prove that a $q$-tuple $\mathbf W_n$ of independent uniformly random cyclically reduced words of length $n$ is \emph{$\lambda$-stable} with probability converging to $1$ exponentially fast. Namely, for every $\Phi\in Aut(F_m)$, the tuple $\Phi(\mathbf W_n)$, after cyclic reduction and symmetrization, satisfies the $C'(\lambda)$ small cancellation condition. Combining generic $\lambda$-stability with Greendlinger normal-closure rigidity and with previous results of Kapovich-Schupp-Shpilrain on generic Nielsen uniqueness and generic Whitehead rigidity we establish, for any fixed $m\ge 2, q\ge 1$, isomorphism rigidity for generic $m$-generator $q$-relator groups. Thus we show that two such generic groups $\langle a_1,\dots, a_m| r_1,\dots, r_q\rangle$ and $\langle a_1,\dots, a_m| s_1,\dots, s_q\rangle$ are isomorphic if and only if, after possibly permuting and inverting the generators $a_1,\dots, a_m$, the relator tuples $(r_1,\dots, r_q)$ and $(s_1,\dots, s_q)$ are the same, up to reordering, cyclic permutations and inverting the relators. Among the applications, we obtain a quadratic-time algorithm that generically solves the isomorphism problem for $m$-generator $q$-relator groups, and show that the number of isomorphism types represented by $m$-generator $q$-relator presentations with cyclically reduced relators of length $n$ is asymptotic to \[ \frac{(2m-1)^{qn}}{2^{m+q}m!\,q!\,n^q}. \] The proof of generic $\lambda$-stability relies on the use of geodesic currents and on our deterministic sufficient criterion for a $q$-tuple $\mathbb W$ in $F_m$ to be $\lambda$-stable in terms of the components of $\mathbb W$ being sufficiently projectively close to filling currents.

Following previous work on metric bisectors with respect to the Korányi metric, we study here bisectors in the Heisenberg group $\mathfrak{H}$ equipped with the Carnot–Carathéodory metric $d_{cc}$.

Thompson's group $F$ is the group of all piecewise-linear homeomorphisms of the unit interval whose breakpoints are dyadic and whose slopes are integer powers of $2$. If $H$ is a finitely generated subgroup of a finitely generated group $G$, its distortion measures the difference between the intrinsic word metric of $H$ and the metric induced from $G$. The subgroup is undistorted when these metrics are equivalent. Guba and Sapir asked whether $F$ contains a distorted copy of itself. The same question was also suggested by Brin. We answer this question negatively: every subgroup of $F$ isomorphic to $F$ is undistorted in $F$.

Given a finite rank Coxeter system, we construct a hyperbolic space by coning off the wide parabolic subcomplexes of its Davis complex. The resulting space is “stability recognizing”, in the sense that stable subspaces of the Davis complex are exactly the quasigeodesically connected subspaces whose images in the coned-off space are quasiisometrically embedded. Equivalently, the coned-off space is “Morse recognizing”: a quasigeodesic in the Davis complex is Morse if and only if it is quasigeodesic in the coned-off space. Similar results are true for the mapping class group of a hyperbolic surface acting on its curve graph. We strengthen this analogy with additional results. First, the action of the Coxeter group on the coned-off space is acylindrical and gives the largest acylindrically hyperbolic structure for the Coxeter group. Second, the stable subgroups of a Coxeter group are characterized as the finitely generated subgroups that are undistorted and purely loxodromic with respect to the action on the coned-off space. The coned-off space, being constructed directly from the Coxeter structure, facilitates concrete characterizations of Morseness and stability for two important Coxeter-native families of objects: combinatorial geodesics and special subgroups.

We characterize the simplicity of reduced crossed product C*-algebras in terms of stabilizer subgroups. Specifically, we prove that if $G$ is a countable group and $X$ is a minimal $G$-flow, then the reduced crossed product C*-algebra $\mathrm{C}(X) \times_\lambda G$ is simple if and only if there is a point in $X$ with a C*-simple stabilizer subgroup. Further, these conditions are equivalent to a generic point in $X$ having a C*-simple stabilizer subgroup. We also provide an example demonstrating that this result does not extend to uncountable groups. This completely resolves a question of Ozawa.

Let $H$ and $K$ be subgroups of a virtually polycyclic group $G$. We prove the Hirsch length inequality $$h(H)+h(K) \leq h(H\cap K)+h(G).$$ We show that equality holds when the number of $(H,K)$-double cosets is finite, and that the converse holds when $G$ is nilpotent. For homomorphisms $\varphi,\psi \colon G \to H$ between virtually polycyclic groups, the equivalent formulation in terms of their coincidence subgroup is $$h(\mathrm{Coin}(\varphi,\psi)) \geq h(G)-h(H).$$ In this formulation, equality holds when the Reidemeister number $R(\varphi,\psi)$ is finite, and the converse holds when $H$ is nilpotent.

We obtain arithmetic conditions which are satisfied by solvable groups admitting an abelian $\pi$-subgroup. First, we extend a previous characterization by removing the square-free hypothesis on some fixed irreducible character degree. We then show the same arithmetic conditions follow from a faithful action of an abelian $\pi$-subgroup on the $\pi'$-part of the Fitting subgroup, where $\pi$ is the set of prime divisors of the aforementioned character degree.

The first singular homology of a Peano continuum $X$ with torsion-free first Cech homology, $\check H_1(x)$, splits as $H_1(X) = \check H_1(X) \oplus K$ where $K$ is the homology shape kernel of $X$. Consequently if a Peano continuum $X$ is a subspace of $\mathbb R^3$, then $H_1(X) = \mathbb Z^\lambda \oplus K$ where $K$ is the homology shape kernel of $X$ and $\lambda$ is a countable cardinal. In the process we construct cotorsion quotients of subgroups of the first homology which correspond to path-connected fibrations of $X$.

We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphism groups, and provides numerous examples of graphing automorphism groups carrying no compatible compact topology, answering a question of Lovasz'. Another consequence of discussion and ensuing constructions is the existence of large mutually locally-globally equivalent graphing families with highly variable symmetry.

Let $L$ be a finite-dimensional Lie algebra over a field $F$. The comaximal graph $\Gamma(L)$ has as vertices the proper nonzero subalgebras of $L$, two of them adjacent whenever they generate $L$; its structure was previously classified for Lie algebras of dimension at most 3 over finite fields. Here we extend that work in two directions. First, we obtain explicit formulas for the number of triangles $t(\Gamma(L))$ for every three-dimensional Lie algebra over $\F_q$. Second, we extend the classification to several four-dimensional families over $\mathbb{F}_q$, the abelian, Heisenberg, and filiform algebras, and $\mathfrak{gl}_2(\F_q)$. We also relate graph-theoretic properties of $\Gamma(L)$, such as completeness and the role of the Frattini subalgebra, to structural properties of $L$, including supersolvability. These results yield new combinatorial invariants for finite-dimensional Lie algebras over finite fields.

It is an open problem to determine whether surface groups can embed in Thompson's group $V$. We prove that any finitely generated subgroup of Thompson's group $T$ without abelian groups of arbitrary high rank is either virtually abelian or virtually free. In particular, surface groups don't embed in $T$, answering a question of Belk and Moore.

Let $G$ be a finite group. The enhanced power graph of $G$, denoted by $\mathcal{E}(G)$, is the graph with vertex set $G$ in which two vertices $u$ and $v$ are adjacent if and only if there exists an element $w \in G$ such that both $u$ and $v$ belong to $\langle w \rangle$. The proper enhanced power graph of $G$, denoted by $\mathcal{E}^{**}(G)$, is the subgraph of $\mathcal{E}(G)$ induced by the non-dominating vertices. The main objective of this paper is to investigate finite groups whose proper enhanced power graph is claw-free, that is, contains no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. We first prove that $\mathcal{E}(G)$ is claw-free if and only if $G$ is cyclic. The set of dominating vertices of $\mathcal{E}(G)$ forms a cyclic subgroup of the center of $G$, namely the cyclicizer $\cyc(G)$ of $G$. This allows us to give a precise description of the structure of $G/\cyc(G)$ when $\mathcal{E}^{**}(G)$ is claw-free. If $G$ is solvable but not nilpotent, then $G$ is metacyclic, or $G/\cyc(G)$ is either a Frobenius group or a $2$-Frobenius group. If $G$ is non-solvable, then $G/\cyc(G)$ is isomorphic to $\PSL(2,q)$ or $\PGL(2,q),$ and this allows us to give a complete classification of the non-solvable groups whose proper enhanced power graph is claw-free.

We construct a new flag-transitive $2$-$(496,4,1)$ design with automorphism group $\mathrm{P\Gamma L}_2(32)$. This corrects an omission in the classification of the finite flag-transitive linear spaces.