Volume 28, Issue 4

Any finitely generated group 𝐺 acts on its asymptotic cones in natural ways. The purpose of this paper is to calculate the kernel of such actions. First, we show that, when 𝐺 is acylindrically hyperbolic, the kernel of the natural action on every asymptotic cone coincides with the unique maximal finite normal subgroup K ⁢ ( G ) K(G) of 𝐺. Secondly, we use this equivalence to interpret the kernel of the actions on asymptotic cones as the kernel of the actions on many spaces at “infinity”. For instance, if G ↷ M G\curvearrowright M is a non-elementary convergence group, then we show that the kernel of actions on the limit set L ⁢ ( G ) L(G) coincides with the kernel of the action on asymptotic cones. Similar results can also be established for the non-trivial Floyd boundary and the CAT ⁡ ( 0 ) \operatorname{CAT}(0) groups with the visual boundary, contracting boundary, and sublinearly Morse boundary. Additionally, the results are extended to another action on asymptotic cones, called Paulin’s construction. In the last section, we calculate the kernel on asymptotic cones for various groups, and as an application, we show that the cardinality of the kernel can determine whether the group admits a non-elementary action under some mild assumptions.

The present paper contributes to the study of asymptotically rigid mapping class groups of infinitely punctured surfaces obtained by thickening planar trees. In a paper from 2022, Genevois, Lonjou and Urech study the latter groups using cube complexes. We determine in which cases their cube complexes are CAT ⁢ ( 0 ) \operatorname{CAT(0)} . From this study, we develop a family of CAT ⁢ ( 0 ) \operatorname{CAT(0)} cubes complexes on which the asymptotically rigid mapping class groups act.

In a paper from 2011, Jiang, Wang and Zhang studied the fixed points and fixed subgroups of selfmaps on a connected finite graph or a connected compact hyperbolic surface 𝑋. In particular, for any selfmap f : X → X f\colon X\to X , they proved that a certain quantity defined in terms of the characteristic chr ⁡ ( f , F ) \operatorname{chr}(f,\mathbf{F}) and the index ind ⁡ ( f , F ) \operatorname{ind}(f,\mathbf{F}) of a fixed point class 𝐅 of 𝑓 is bounded below by 2 ⁢ χ ⁢ ( X ) 2\chi(X) , where χ ⁢ ( X ) \chi(X) is the Euler characteristic of 𝑋. In this paper, we give a sufficient condition for when equality holds and hence we partially answer a question of Jiang. We do this by studying iwip outer endomorphisms of free groups acting on stable trees.

The goal of this paper is to show the following result: for every integer n ≥ 2 n\geq 2 , there is a countable orderable group whose space of orders is countable and has Cantor–Bendixson rank 𝑛. We show this by explicitly constructing a family of orderable groups with the desired properties.

Let 𝐾 be a non-archimedean local field. We show that discrete subgroups without 2-torsion in PSL 2 ⁢ ( K ) \mathrm{PSL}_{2}(K) can always be lifted to SL 2 ⁢ ( K ) \mathrm{SL}_{2}(K) , and provide examples (when char ⁡ ( K ) ≠ 2 \operatorname{char}(K)\neq 2 ) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into PSL 2 ⁢ ( K ) \mathrm{PSL}_{2}(K) to SL 2 ⁢ ( K ) \mathrm{SL}_{2}(K) .

In a paper from 2006, Schulte and Weiss initiated the problem of characterizing the regular polytopes of orders 2 n ⁢ p 2^{n}p for 𝑛 a positive integer and 𝑝 an odd prime. In this paper, we first prove that if a 3-polytope of order 2 n ⁢ p 2^{n}p has Schläfli type { k 1 , k 2 } \{k_{1},k_{2}\} , then p ∣ k 1 p\mid k_{1} or p ∣ k 2 p\mid k_{2} . This leads to two classes, up to duality, for the Schläfli type, namely Type (1) where k 1 = 2 s ⁢ p k_{1}=2^{s}p and k 2 = 2 t k_{2}=2^{t} and Type (2) where k 1 = 2 s ⁢ p k_{1}=2^{s}p and k 2 = 2 t ⁢ p k_{2}=2^{t}p . We then show that there exists a regular 3-polytope of order 2 n ⁢ p 2^{n}p with Type (1) when s ≥ 2 s\geq 2 , t ≥ 2 t\geq 2 and n ≥ s + t + 1 n\geq s+t+1 coming from a general construction of regular 3-polytopes of order 2 n ⁢ ℓ 1 ⁢ ℓ 2 2^{n}\ell_{1}\ell_{2} with Schläfli type { 2 s ⁢ ℓ 1 , 2 t ⁢ ℓ 2 } \{2^{s}\ell_{1},2^{t}\ell_{2}\} , where both ℓ 1 \ell_{1} and ℓ 2 \ell_{2} are odd. Furthermore, for p = 3 p=3 and n ≥ 7 n\geq 7 , we show that there exists a regular 3-polytope of order 3 ⋅ 2 n 3\cdot 2^{n} with type { 6 , 2 s } \{6,2^{s}\} if and only if 2 ≤ s ≤ n − 2 2\leq s\leq n-2 and s ≠ n − 3 s\neq n-3 . For Type (2), we prove that there exists a regular 3-polytope of order 2 n ⋅ 3 2^{n}\cdot 3 with Schläfli type { 6 , 6 } \{6,6\} when n ≥ 5 n\geq 5 coming from a general construction of regular 3-polytopes of Schläfli type { 6 , 6 } \{6,6\} with orders 192 ⁢ m 3 192m^{3} , 384 ⁢ m 3 384m^{3} or 768 ⁢ m 3 768m^{3} for any positive integer 𝑚.

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation x n = 1 x^{n}=1 in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer equations than unknowns. Instead of systems of equations, we consider arbitrary first-order formulae in the group language with constants. Our result substantially generalizes the Klyachko–Mkrtchyan theorem (2014) on this topic.

We define and investigate the property of being “exponent-critical” for a finite group. A finite group is said to be exponent-critical if its exponent is not the least common multiple of the exponents of its proper non-abelian subgroups. We explore properties of exponent-critical groups and give a characterization of such groups. This characterization generalizes a classical result of Miller and Moreno on minimal non-abelian groups; interesting families of 𝑝-groups appear.

The article deals with finite groups in which commutators have prime power order (CPPO-groups). We show that if 𝐺 is a soluble CPPO-group, then the order of the commutator subgroup G ′ G^{\prime} is divisible by at most two primes. This improves an earlier result saying that the order of the commutator subgroup G ′ G^{\prime} is divisible by at most three primes.

For a finite group 𝐺, set Irr N ⁡ ( G ) := { χ ∈ Irr ( G ) ∣ χ ( 1 ) 2 ∤ | G : ker χ | } , cd N ⁡ ( G ) := { χ ⁢ ( 1 ) ∣ χ ∈ Irr N ⁡ ( G ) } . \begin{aligned} \operatorname{Irr}_{N}(G)&:=\{\chi\in\operatorname{Irr}(G)\mid\chi{(1)^{2}}\nmid\lvert G:\operatorname{ker}\chi\rvert\},\\ \operatorname{cd}_{N}(G)&:=\{\chi(1)\mid\chi\in\operatorname{Irr}_{N}(G)\}.\end{aligned} In this paper, we study finite groups 𝐺 with | cd N ⁡ ( G ) | ≤ 1 \lvert\operatorname{cd}_{N}(G)\rvert\leq 1 . In particular, we prove that if 𝐺 has odd order, then h ⁢ ( G ) ≤ | cd N ⁡ ( G ) | + 1 h(G)\leq\lvert\operatorname{cd}_{N}(G)\rvert+1 , where h ⁢ ( G ) h(G) is the Fitting height of 𝐺.

In this paper, we prove that the Hilbert divisors of irreducible Brauer characters in 2-blocks with nontrivial abelian defect groups are strictly greater than 1. This confirms a conjecture of Liu and Willems in this case. The proof relates the conjecture with a problem of Feit, which asks if the 𝑝-part of the degree of an irreducible Brauer character 𝜙 of 𝐺 is always less than the 𝑝-part of the order of 𝐺. We resolve Feit’s problem positively for 2-blocks with abelian defect groups. But it is well known that the question has a negative answer for 2-blocks with non-abelian defect groups.

Given a character triple ( G , N , θ ) (G,N,\theta) , which means that 𝐺 is a finite group with N ⁢ ⊲ ⁢ G N\vartriangleleft G and θ ∈ Irr ⁡ ( N ) \theta\in\operatorname{Irr}(N) is 𝐺-invariant, we introduce the notion of a 𝜋-quasi-extension of 𝜃 to 𝐺, where 𝜋 is the set of primes dividing the order of the cohomology element [ θ ] G / N ∈ H 2 ⁢ ( G / N , C × ) [\theta]_{G/N}\in H^{2}(G/N,\mathbb{C}^{\times}) associated with the character triple, and then we establish the uniqueness of such an extension in the normalized case. As an application, we use the 𝜋-quasi-extension of 𝜃 to construct a bijection from the set of 𝜋-defect zero characters of G / N G/N onto the set of relative 𝜋-defect zero characters of 𝐺 over 𝜃. Our results generalize the related theorems of M. Murai and of G. Navarro.