KyeongRo Kim
Calegari and Loukidou introduced zippers, consisting of a disjoint pair of invariant real trees in the boundary of a closed hyperbolic 3-manifold group $π_1(M)$, which ensure the existence of a universal circle. We study the action of $π_1(M)$ on a minimal zipper and prove a fixed point dichotomy: every nontrivial element either fixes a unique point in each tree or acts freely on both. This answers a question of Calegari and Loukidou. As a consequence, there exists an element with exactly one fixed point in each tree.
Christopher P. Bendel, Daniel K. Nakano, Cornelius Pillen
Let $G$ be a reductive algebraic group scheme defined over ${\mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p$. There are two associated families of finite group schemes, the $r$-th Frobenius kernels, denoted by $G_r$, and the fixed points of the iterated Frobenius map, the finite groups of Lie type, denoted by $G(\mathbb{F}_q).$ Bendel, Nakano and Pillen initiated the investigation of the induction functor $\operatorname{ind}_{G(\mathbb{F}_q)}^G-$. Using filtrations and truncation, large amounts of data coming from the algebraic group and the Frobenius kernels can be transferred to the finite group. This paper looks at connections between a fundamental theorem of Chastkofsky and Jantzen and the induction functor via the cohomology and representation theory of $G$.
Benjamin Klopsch, Margherita Piccolo, Britta Späth
The representation zeta function of a profinite group $G$ encodes the distribution of continuous irreducible complex representations of $G$ as a function of the dimension. Its abscissa of convergence $α(G)$ describes the polynomial degree of representation growth of $G$. Within the class of quasi-semisimple profinite groups, we characterise those of polynomial representation growth (PRG) and we prove that whether such a group $G$ has PRG or not only depends on its semisimple part $G/\mathrm{Z}(G)$. Moreover, we show that, for quasi-semisimple profinite groups $G$ that have uniformly bounded Lie ranks, the degree of growth satisfies $α(G) = α(G/\mathrm{Z}(G))$. We provide a technique to produce, for any prescribed positive real number $\varrho$, quasi-semisimple profinite groups $G$ with PRG of degree $α(G) = \varrho$. Our method allows for considerable flexibility regarding the inclusion of finite simple groups of Lie type as composition factors of $G$. Furthermore, we can arrange for the groups $G$ of prescribed representation growth to be profinite completions of suitable finitely generated discrete groups $Γ$ so that the group $Γ$ has the same representation zeta function as $G$.
Marco Praderio Bova
We develop tools which use common fusion systems building techniques in order to compute higher limits over the centric orbit category. We apply these tools in order to study both the Diaz-Park sharpness conjecture as well as the weaker cohomological sharpness conjecture which predicts vanishing of higher limits only for the cohomology Mackey functors . Our approach leads to proving cohomological sharpness (but not sharpness) for all saturated fusion systems over p-groups of either maximal nihlpotency or of rank 2 and all polynomial, Henke-Shpectorov and van Beek fusion systems. This list includes all but 2 of the cases for which cohomological sharpness was previously known as well as most currently known families of exotic fusion systems. For the polynomial, Henke-Shpectorov and 6 of the van Beek fusion systems, sharpness is also approximated by proving vanishing of all but the first higher limits of any Mackey functor. The distinction our approach makes between sharpness and cohomological sharpness is somewhat surprising and interesting by itself. Our approach draws a new connection between cohomological sharpness and fusion system building techniques. We believe that this connection will lead to a better understanding of both fusion systems and Mackey functors over them.
Lvzhou Chen, Nicolaus Heuer
We show a uniform spectral gap of stable commutator length for all compact hyperbolic $2$-orbifolds relative to the peripheral subgroups. Except for the case of a sphere with three cone points, we have an explicit uniform gap $1/36$. These estimates are needed in understanding stable commutator length in $3$-manifolds. Our methods use explicit quasimorphisms for the generic case, and use hyperbolic geometry (pleated surfaces) for the exceptional case of a sphere with three cone points.
Steven Amelotte, Vladimir Gorchakov
We construct models for the classifying spaces of coabelian subgroups of right-angled Coxeter groups as homotopy orbit spaces of real moment-angle complexes, generalizing well-known models for the classifying space of a right-angled Coxeter group and its commutator subgroup. This identifies the cohomology of these groups with the Borel equivariant cohomology of elementary abelian $2$-group actions on cubical subcomplexes of a cube $[-1,1]^m$. We then characterize equivariant formality for these actions, leading to a simple graph-theoretic criterion for when the cohomology of a coabelian subgroup is free as a module over the cohomology of the quotient by the commutator subgroup of the right-angled Coxeter group.
Daria Poliakova
We study cubic realizations of posets compatible with projection maps, meaning that the projection is represented by deletion of the last coordinate. For cylindrical projections, we introduce the pre-Reeb graph and the augmented pre-Reeb graph, which control compatible cubic lifts and compatible order-embedding cubic lifts, respectively. We apply this construction to the deletion towers in weak order of types \(A\) and \(B\). The pre-Reeb graphs are the \(1\)-skeleta of, respectively, cubes and certain zonotopes. In both cases, the augmented pre-Reeb graphs have reachability posets that are total orders, yielding combinatorial uniqueness of the compatible order-embedding cubic coordinates.
Martin Bachratý
A skew morphism of a finite group $B$ is a permutation of $B$ fixing the identity and satisfying $\varphi(xy) = \varphi(x)\varphi^{i_x}(y)$ for some integers $i_x$ indexed by $x \in B$. The enumeration of skew morphisms of finite cyclic groups remains an open problem. The most substantial progress to date concerns cyclic $p$-groups with $p$ odd, for which a full classification and enumeration was obtained by Kovács and Nedela. In this paper we treat the remaining case $p = 2$, giving a complete classification and enumeration of skew morphisms of finite cyclic $2$-groups. Writing $\mathrm{Skew}(n)$ for the number of skew morphisms of $\mathbb{Z}_n$, we prove that $\mathrm{Skew}(2^e) = 4\,\mathrm{Skew}(2^{e-1}) - 4$ for each $e \geq 4$, and that $\mathrm{Skew}(2^e) = (7 \cdot 4^{e-2} + 8)/6$ for each $e \geq 3$. This completes the enumeration of skew morphisms for all cyclic $p$-groups.
Piotr Kowalski, Pınar Uğurlu Kowalski
We show that generic automorphisms of stable groups are supertight in a strong sense. In particular, we obtain the existence of supertight automorphisms. We also answer a question concerning the relationship between supertight automorphisms of $\mathrm{PGL}_2(K)$ and generic automorphisms of the underlying field $K$. Moreover, we provide partial evidence-already suggested by Hrushovski-toward the principle that ``fixed points are pseudofinite'' in the setting of generic automorphisms of simple groups of finite Morley rank.
Francis Brown
Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for multivariable Vandermonde determinants as a sum of completely factorising terms, each of which is a Vandermonde determinant in fewer variables. As an application, we deduce an elementary proof of the multiplicativity of the transfinite diameter for products of compact sets.
Gernot Stroth
We consider saturated fusion systems $\mathcal F$ on a Sylow $2$-subgroup of $Ω^+_8(2)$ with $O_2(\mathcal F) = 1$. Examples for this are the $2$-fusion systems of $Ω^+_8(2)$, $Ω^+_8(2):3$, $PΩ^+_8(3)$ and $PΩ^+_8(3):3$
Kun Zhang, Yuanyang Zhou
In this paper, with suitable assumptions, we generalize the work of Külshammer and Puig on extensions of nilpotent blocks to inertial blocks.
Guillaume Dumas, Jingyin Huang, Srivatsav Kunnawalkam Elayavalli, Lizzy Teryoshin
The number of connected components can be remembered by the von Neumann algebra among Artin groups, the only possible exception being the case that corresponds to the free group factor problem. In the case of Coxeter groups, this result is obtained in the absence of relatively hyperbolicity. We also discuss a specific case of the analogous problem in measure equivalence where each factor group is a product of nonabelian free groups.
Yue Xin, Yan Li, Bingzhe Hou
In this paper, we study the right invariant metric $d_{H^{\infty}}$ on the analytic automorphism group $\rm{Aut}(\mathbb{D})$ of the unit open disk $\mathbb{D}$ induced by maximal modulus, that is, $d_{H^{\infty}}(\varphi, ψ)=\sup_{z\in\mathbb{D}}|\varphi(z)-ψ(z)|$ for any $\varphi, ψ\in \rm{Aut}(\mathbb{D})$. We give the explicit formula of the right invariant metric $d_{H^{\infty}}$ and characterize the almost regular Finsler geometric structure of $(\rm{Aut}(\mathbb{D}), d_{H^{\infty}})$.
Mohamad N. Nasser, Oscar Ocampo
We introduce linear representations of the universal virtual braid group $UV_n(c)$, where $n\geq 2$ and $c\geq 1$, which is a unifying framework for braid-type groups with multiple types of crossings. We classify and study its complex homogeneous $2$-local representations for all $n\geq 3$ and $c\geq 1$ (unique up to equivalence) and complex homogeneous $3$-local representations for all $n\geq 4$ and $c=2$ (four distinct families). We then introduce the universal welded braid group $UW_n(c)$ as a quotient of $UV_n(c)$ by the welded relations. This group recovers all known welded-type groups as quotients. We prove that $UW_n(c)$ has abelianization $\mathbb{Z}^c \oplus \mathbb{Z}_2$, perfect commutator subgroup for $n \geq 5$, trivial center, and $S_n$ as its smallest non-abelian finite quotient. Finally, we classify and study the complex homogeneous $2$-local representations of $UW_n(c)$ for all $n\geq 3$ and $c\geq 1$, obtaining three distinct families.
A. Ballester-Bolinches, R. Esteban-Romero, P. Jiménez-Seral, V. Pérez-Calabuig
In our previous work: Adv. Math. 455 (2024), no. 109880, solubility of solutions was introduced as an extension of solubility of skew braces in the classification context of non-degenerate solutions of the Yang-Baxter equation. One of our main results (Theorem C) proved that a skew brace is soluble if, and only if, its associated solution is soluble. A minor step depending on the definition of homomorphism of solutions was overlooked. In this work, proof of Theorem C is repaired by means of a new class of homomorphisms of solutions: i-homomorphisms of solutions. The importance of this new class is twofold: indecomposable solutions are characterised by means of i-simplicity of solutions, and i-kernels of i-homomorphisms generate ideals in structure skew braces of solutions. Hence, solubility of solutions is redefined as an opposite class of indecomposable solutions. The results obtained with this definition improve our previous outcomes: every soluble solution is proved to have a soluble structure skew brace, and consequently, Theorem C still holds. Several results stemming from this new analysis are outlined.
Kıvanç Ersoy
We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $ω$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group. On the other hand, we show that locally graded non-sofic groups are necessarily $ω$-non-sofic. More precisely, such groups contain finitely generated non-sofic subgroups admitting strictly decreasing chains of finitely generated normal subgroups whose intersection is nontrivial and lies in the profinite residual. Finally, using results on existentially closed groups, we prove that the existence of a non-sofic group implies the existence of a countable existentially closed non-sofic group whose centralizers form a densely ordered chain of non-sofic subgroups of order type $(\mathbb{Q},\leq)$. In particular, we show that if a non-sofic group exists, then the class of $ω$-non-sofic groups is non-empty. Moreover, we prove that the existence of a non-sofic group implies the existence of a non-sofic group of unbounded exponent.
SK Kiran Ajij
Bestvina-Feighn-Handel show that for finitely many generic and independent hyperbolic automorphisms $φ_1, \cdots, φ_r$ of $F_n$, the resulting extension $F_n \rtimes F_r$ is hyperbolic. This paper generalizes the above statement to the case where $φ_1, \cdots, φ_r$ are hyperbolic non-surjective endomorphisms of $F_n$. In our case the output is a multiple HNN extension associated to a graph with one vertex and $r$ edges. All edge and vertex groups are isomorphic to $F_n$.
Wouter van Doorn, Pietro Monticone, Quanyu Tang
For a family $(A_q)_{q\in Q}$ of subsets of a semigroup, the product intersection set records those exponents $h \in \mathbb{N}$ for which the $h$-fold product set of the intersection, $(\bigcap_q A_q)^h$, is equal to $\bigcap_q A_q^h$, the intersection of the product sets. Nathanson recently asked which subsets of $\mathbb{N}$ can occur as a product intersection set, both for arbitrary and for decreasing families $(A_q)_{q\in Q}$. We solve both problems by giving a complete classification. In particular, when $|Q| \ge 2$, we show that in either case any subset $X \subseteq \mathbb{N}$ with $1 \in X$ occurs as a product intersection set. Both classifications were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.
Fabricio Dos Santos, Christophe Hohlweg, Aleksandr Trufanov
In this article, we propose to initiate the general study of involution systems. An {\em involution system}, that is, a group $W$ generated by a set of involutions $S$, is naturally endowed with a {\em weak order} arising from orienting the Cayley graph of $(W,S)$. In the case of a Coxeter system $(W,S)$, Björner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their related structures. In this article, we discuss the following question: For which involution systems is the weak order a complete meet-semilattice? The class of involution systems that satisfies this condition is larger than the class of Coxeter systems (it contains, for instance, Cactus groups). In the case of an involution system with sign character, we provide a finite presentation by generators and relations and a classification in rank 3. We also obtain new characterizations of Coxeter systems in terms of the weak order, and prove a number of results on certain subclasses of these involution systems. Finally, we discuss further works and open problems in relation to biautomatic structures, geometric representations, mediangle graphs, and more.