Band 26, Heft 4

We describe a method, based on curvature distribution techniques on van Kampen diagrams, for proving finitely presented groups hyperbolic. We apply our method and show that the generalised Fibonacci group F ⁢ ( r , n ) F(r,n) is hyperbolic when r ≥ 3 r\geq 3 and n ≥ 6 ⁢ r + 1 n\geq 6r+1 and determine which of the groups F ⁢ ( 3 , n ) F(3,n) are hyperbolic.

One version of Whitehead’s famous cut vertex lemma says that if an element of a free group is part of a free basis, then a certain graph associated to its conjugacy class that we call the star graph is either disconnected or has a cut vertex. We state and prove a version of this lemma for conjugacy classes of elements and convex-cocompact subgroups of groups acting cocompactly on trees with finitely generated edge stabilizers.

We give a characterization of finitely presented Bestvina–Brady groups which split over abelian subgroups and describe the JSJ-decompositions of those Bestvina–Brady groups.

We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated, and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.

We say a group 𝐺 has property R ∞ R_{\infty} if the number R ⁢ ( φ ) R(\varphi) of twisted conjugacy classes is infinite for every automorphism 𝜑 of 𝐺. For such groups, the R ∞ R_{\infty} -nilpotency degree is the least integer 𝑐 such that G / γ c + 1 ⁢ ( G ) G/\gamma_{c+1}(G) has property R ∞ R_{\infty} . In this work, we compute the R ∞ R_{\infty} -nilpotency degree of all Generalized Solvable Baumslag–Solitar groups Γ n \Gamma_{n} . Moreover, we compute the lower central series of Γ n \Gamma_{n} , write the nilpotent quotients Γ n , c = Γ n / γ c + 1 ⁢ ( Γ n ) \Gamma_{n,c}=\Gamma_{n}/\gamma_{c+1}(\Gamma_{n}) as semidirect products of finitely generated abelian groups and classify which invertible integer matrices can be extended to automorphisms of Γ n , c \Gamma_{n,c} .

Answering Questions 19.23 and 19.24 from the Kourovka notebook, we construct polycyclic groups with arbitrary torsion lengths and give examples of finitely presented groups whose quotients by their torsion subgroups are not finitely presented.

Suppose that 𝒞 is a class of groups consisting only of periodic groups and P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} is the set of prime numbers that do not divide the order of any element of a 𝒞-group. It is easy to see that if a subgroup 𝑌 of a group 𝑋 is 𝒞-separable in this group, then it is P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated in 𝑋. Let us say that 𝑋 has the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} if all of its P ⁢ ( C ) ′ \mathfrak{P}(\mathcal{C})^{\prime} -isolated subgroups are 𝒞-separable. We find a condition that is sufficient for a nilpotent group 𝑁 to have the property C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} provided 𝒞 is a root class. We also prove that if 𝑁 is torsion-free, then the indicated condition is necessary for this group to have C ⁢ - ⁢ S ⁢ e ⁢ p \mathcal{C}\textup{-}\mathfrak{Sep} .

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for abelian groups, Trans. Amer. Math. Soc. 361 (2009), 7, 3401–3434]. Later, this result was extended to all abelian groups [D. Dikranjan and A. Giordano Bruno, Entropy on abelian groups, Adv. Math. 298 (2016), 612–653] and, recently, to all torsion finitely quasihamiltonian groups [A. Giordano Bruno and F. Salizzoni, Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups, J. Group Theory 23 (2020), 5, 831–846]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [A. Giordano Bruno and P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory 20 (2017), 4, 763–774]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.

We prove that if 𝐺 is a connected semisimple algebraic group of rank 𝑟, and 𝐻 is a subgroup of 𝐺 that is contained in no proper parabolic subgroup, then we have | C G ⁢ ( H ) | < c r ⁢ | Z ⁢ ( G ) | \lvert C_{G}(H)\rvert<c^{r}\lvert Z(G)\rvert , where 𝑐 is an absolute constant ( c = 16 c=16 if all simple factors of 𝐺 are classical, and c ≤ 197 c\leq 197 in general).

We prove that the order of a finite group 𝐺 with trivial solvable radical is bounded above in terms of acd ⁡ ( G ) \operatorname{acd}(G) , the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of acd ⁡ ( G ) \operatorname{acd}(G) , but we show that, in certain cases, it is bounded in terms of the degrees of the irreducible characters of 𝐺 that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck’s conjecture.

A finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.

Let 𝐺 be a group. A function G → G G\to G of the form x ↦ x α ⁢ g x\mapsto x^{\alpha}g for a fixed automorphism 𝛼 of 𝐺 and a fixed g ∈ G g\in G is called an affine map of 𝐺 . The affine maps of 𝐺 form a group, called the holomorph of 𝐺. In this paper, we study finite groups 𝐺 with an affine map of large order. More precisely, we show that if 𝐺 admits an affine map of order larger than 1 2 ⁢ | G | \frac{1}{2}\lvert G\rvert , then 𝐺 is solvable of derived length at most 3. We also show that, more generally, for each ρ ∈ ( 0 , 1 ] \rho\in(0,1] , if 𝐺 admits an affine map of order at least ρ ⁢ | G | \rho\lvert G\rvert , then the largest solvable normal subgroup of 𝐺 has derived length at most 4 ⁢ ⌊ log 2 ⁡ ( ρ − 1 ) ⌋ + 3 4\lfloor\log_{2}(\rho^{-1})\rfloor+3 .

In recent work, Cameron, Manna and Mehatari have studied the finite groups whose power graph is a cograph, which we refer to as power-cograph groups. They classify the nilpotent groups with this property, and they establish partial results in the general setting, highlighting certain number-theoretic difficulties that arise for the simple groups of the form PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . In this paper, we prove that these number-theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 ⁡ ( q ) \operatorname{PSL}_{2}(q) or Sz ⁡ ( 2 2 ⁢ e + 1 ) \operatorname{Sz}(2^{2e+1}) . For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg–Kegel graph is connected. Moreover, we reduce the case where the Gruenberg–Kegel graph is disconnected to the classification of 𝑝-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.