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Wednesday, 4 November 2026: 6 new papers, 8 cross-lists and 8 replacements in Group Theory (math.GR), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
An invariant random subgroup of a discrete group is a conjugation-invariant probability measure on the space of its subgroups. We confirm a conjecture of Simon Thomas and classify the invariant random subgroups of the finitary special and general linear groups over a finite field. These are shown to be stabilizers of random tuples of vectors and functionals. The classification uses the character theory of the finitary special linear group established by Skudlarek and the theory of the invariant random subgroups of the finitary permutation group established by Vershik. In addition, we classify the invariant probability measures for several natural actions of these finitary groups.
Let $F$ be a finitely generated free group. We prove that the centraliser of any element of $\mathrm{Out}(F)$ or $\mathrm{Aut}(F)$ is of type $\rm{VF}$. We also prove that the (outer) automorphism group of any free-by-cyclic group $M=F\rtimes_\varphi\mathbb{Z}$ is of type $\rm{VF}$, and so are the McCool subgroups of $\mathrm{Out}(M)$. This relies on the construction of a new $\mathrm{Out}(M)$-invariant splitting of $M$, that enables us to reduce the proof of the above statements to the case where the outer class of $\varphi$ has finite order.
Do Kaplansky's group-ring conjectures hold for torsion-free word-hyperbolic groups? We show that all four fail in positive characteristic. For every prime $p$ there is a finitely presented, torsion-free, word-hyperbolic group $G$ with a finite two-dimensional classifying space and $a,b\in\mathbb{F}_p[G]$ with $ab=1\neq ba$. So, over every field of characteristic $p$, the group algebra of $G$ is not directly finite and has a non-trivial idempotent, zero divisor and unit, and $G$ is neither sofic nor surjunctive. For $p=2$, $G$, $a$ and $b$ come from OpenAI's random construction, and we add that $G$ is hyperbolic with probability $1-o(1)$, as are OpenAI's zero-divisor groups. For odd $p$ the groups are new: they adapt OpenAI's direct-finiteness construction with a signed cancellation, found first by S. V. Ivanov and independently by us, via an exact solution of a colour-balance problem and a Perron certificate. As far as we know, they are the first torsion-free counterexamples to direct finiteness and to the idempotent conjecture in odd characteristic. Hyperbolicity comes from one deterministic criterion: if a labelled graph is pattern-free, the group obtained by coning off its components is aspherical and has a linear isoperimetric inequality. We prove it by running OpenAI's exclusion of reduced spheres on van Kampen discs, an accounting OpenAI used for another complex. For every prime $p$, the existence of $G$, $a$ and $b$ and the failure of all four conjectures are formalized in Lean 4, including existence forms of the random-graph estimates used. In characteristic zero, where the idempotent and direct-finiteness conjectures hold for torsion-free hyperbolic groups, we show that no cancellation built into the vertex types survives the decay estimate that makes the groups torsion-free. OpenAI's characteristic-two counterexample with torsion is a CAT(0) group.
The \emph{enhanced power graph} $\PE(G)$ of a finite group $G$ has vertex set $G$, two elements being adjacent if and only if they generate a cyclic subgroup. We determine the finite simple groups $G$ for which $\PE(G)$ is chordal: they are the cyclic groups of prime order, $\Alt(7)$, $\PSL_2(q)$ ($q\ge4$), $\Sz(q)$ ($q=2^{2n+1}\ge8$) and $\PSL_3(4)$. This answers an open problem of Bubboloni, Fumagalli and Praeger \cite{BFP}. We also prove that for every finite group $G$, if the power graph $P(G)$ is a cograph (resp.\ chordal) then so is $\PE(G)$; the cograph case is a conjecture of Bubboloni, Fumagalli and Praeger \cite{BFP}.
We propose a new refinement of the (blockwise) Alperin weight conjecture taking dimensions of simple modules (or degrees of irreducible Brauer characters) into consideration. This is similar to Giannelli's refinement of the (Alperin--)McKay conjecture. We prove this new refinement for blocks with a cyclic or Klein four defect group, for inertial blocks (which include the Brou\'{e}--Puig nilpotent blocks), and for $p$-blocks of $p$-solvable groups.
In this paper we begin developing a theory of Morse hierarchies for finitely generated groups. Very roughly, a Morse hierarchy is a iterative splitting of a group over subgroups that do not contribute to the Morse boundary of the ambient group. We apply this theory to Coxeter groups, establishing new combinatorial criteria that distinguish certain Coxeter groups from other Coxeter groups as well as right-angled Artin groups up to quasi-isometry. We highlight the consequences for the right-angled case. To build the reader's intuition, the last section provides many examples of Morse hierarchies and their subtleties. Finally, there is an appendix by the second author and Vivian He that extends these ideas to the metrizable Cashen--Mackay topology on the Morse boundary.
We present discrete, faithful and purely loxodromic representations of a closed surface group of genus 431, and of a free group, into the group of isometries of a finite-dimensional real-hyperbolic space for which no Cannon--Thurston map exists. The construction combines three ingredients: the small-cancellation strategy employed by Baker--Riley to construct a subgroup of an ambient hyperbolic group with no Cannon--Thurston map, the realization of these groups by octagon complexes, and a theorem of Desgroseilliers--Haglund that supplies a convex cocompact faithful representation of the ambient hyperbolic group on a real-hyperbolic space. These examples admit ray Cannon--Thurston maps, as defined by Halder, Mahan Mj and Sardar.
Given a finitely aligned higher-rank $k$-graph $\Lambda$, Farthing, Muhly and Yeend (Semigroup Forum 2005) constructed an inverse semigroup $S_\Lambda$ to provide a groupoid model for the $C^*$-algebra of $\Lambda$. In this paper, we extend some of the existing results related to the semigroup structure of a graph inverse semigroup to the more general setting of the inverse semigroup $S_\Lambda$. We obtain geometric descriptions of the Green's relations on $S_\Lambda$, which effectively generalize the existing characterization of Green's relations on a graph inverse semigroup. We provide a simple and useful description of the Cuntz-Krieger ideal of the Booleanization of $S_\Lambda$ and, based on this, identify a certain $X$-to-join Booleanization of $S_\Lambda$, where the abstract and concrete Cuntz-Krieger relations agree. Besides the natural partial order, $S_\Lambda$ is also ordered by set-inclusion. This helps us to realize $S_\Lambda$ as a semigroup \emph{graded-ordered} by $\mathbb{Z}^k$ and, subsequently, gives rise to a full inverse subsemigroup $S_\Lambda^h$ of $S_\Lambda$, which is graded by $\mathbb{Z}^k$ in the usual sense. We show that the Kumjian-Pask algebra of $\Lambda$ is graded isomorphic to the quotient of the contracted semigroup algebra of $S_\Lambda^h$ by the tight ideal. In the final part of the paper, we show that a $k$-graph morphism $p:\Lambda_1\longrightarrow \Lambda_2$ that is injective and satisfies a certain path lifting property induces, in a natural way, an injective $0$-preserving semigroup homomorphism from $S_{\Lambda_1}$ to $S_{\Lambda_2}$. In view of this, we build a suitable categorical setup to conclude that the isomorphism established between the Kumjian-Pask algebra of $\Lambda$ and the tight ideal quotient of the semigroup algebra of $S_\Lambda^h$ is natural.
This paper is the second in a series about (relatively) free profinite semigroups and S-adic representations of minimal shifts. The first installment introduced the key notions of profinite images and saturating properties. The present paper examines the relationship between relative invertibility conditions (on a directive sequence) and charging properties (of a shift). The main result shows that the two properties are equivalent under mild conditions. Moreover, charging properties are shown to be equivalent to density of subgroups generated by return words in relatively free profinite groups. The main result can then be used to deduce purely discrete statements about subgroups generated by return words in relatively free groups. We also highlight applications of the main result to the study of maximal bifix decoding, following earlier work by the second author. Finally, we establish some results on the effect of direct images on charging properties and explain how to apply these results to the case of episturmian shifts.
We describe an algorithm and a computer program computing the relative Tate-Shafarevich kernel $Sha^S(K,G)$ of a reductive group $G$ over a global field $K$ with respect to a finite set $S$ of places of $K$ containing no archimedean places. We assume that we know the algebraic fundamental group $\Lambda=\pi_1^{alg}(G)$, the Galois group $\Theta=Gal(L/K)$ acting on $\Lambda$, where $L$ is a finite Galois extension of $K$ having no real places, the decomposition subgroups $\Theta_v\subseteq \Theta$ for all places $v$ in $S$, and all non-cyclic decomposition subgroups $\Theta_v$ for $v$ outside $S$. We also compute $Sha^S(K,G)$ in some cases where $S$ contains archimedean places.
A space is feebly compact if every locally finite family of nonempty open subsets of it is finite; a Tychonoff space is feebly compact if and only if it is pseudocompact. M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a $k$-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group. The proof is based on the following result: if $X$ is a separable Baire space, $Y$ is a feebly compact space and $Z$ is a regular space, then every separately continuous map $X\times Y\to Z$ is strongly quasicontinuous. Unlike the known approaches for Tychonoff spaces, the proof does not use real-valued functions. More generally, every regular separable semitopological group which is $\Gamma^{BM}_o$-nonmeager, a Baire-type property defined by a modification of the Banach-Mazur game, is a topological group. This class of groups contains all regular separable Baire groups which are feebly $p$-spaces; feebly $p$-spaces form a wider class of spaces, which contains all feebly compact spaces and all $p$-spaces. The results remain valid for spaces satisfying the assumptions locally. In particular, every regular locally separable locally feebly compact semitopological group is a topological group.
In a weakly integral braided fusion category, a prime divides none of the conjugacy-class dimensions if and only if it does not divide the Frobenius-Perron dimension of the adjoint subcategory. We prove that these conditions hold precisely when the corresponding Sylow subgroup of the group of invertible objects generates a braided Deligne factor whose dimension is the largest power of that prime dividing the Frobenius-Perron dimension of the category. We also prove that the prime graph of the conjugacy-class dimensions has at most two connected components. If the graph is disconnected, the category is solvable. The bound is attained by the representation category of the Drinfeld double of the symmetric group on three letters.
Erd\H{o}s and Gy\'arf\'as conjectured that every graph with minimum degree at least three contains a power-of-two-length cycle. We prove every Cayley graph of odd order satisfies the conjecture. As a corollary, we find that Cayley graphs of finite nilpotent groups satisfy the conjecture as well.
Let $p$ be a prime. We classify homomorphisms $H:V_1\to V_2$ between abelian $p$-groups that are direct sums of cyclic groups, under the assumption $p^2V_2=0$. We prove that every such homomorphism is isomorphic to a direct sum of finite indecomposable representations and that this decomposition is unique up to isomorphism and reordering of the summands.