Recently, Kim and Lee [9] presented an example of a non-Hopfian relatively hyperbolic group with a Hopfian peripheral subgroup, demonstrating a counterexample to Osin's well-known question [12, Problem 5.5]. In this paper, we provide a general construction method using the so-called image extension theorem to generate non-Hopfian relatively hyperbolic groups with Hopfian peripheral subgroups. Additionally, we provide two specific examples using this construction method.
We classify the subalgebras of the real forms of the special linear algebra sl3(C), specifically the real special linear algebra sl3(R), the special unitary algebra su(3), and the generalized special unitary algebra su(2,1). Our method uses Galois cohomology together with the known classification of complex subalgebras of sl3(C). The real subalgebras of sl3(R) have already been classified by Winternitz using different techniques than ours. We recover this classification, and amend minor inaccuracies. However, the classifications of the real subalgebras of su(3), and su(2,1) are new.
We introduce the notion of rigid extensions to generalize a key lemma underlying the Amitsur–Small Theorem to the skew setting. Building on this framework, we establish a skew version of the Amitsur–Small Theorem.
We show that the higher direct images of smooth commutative group schemes from the Kummer log flat site to the classical flat site are torsion. For cases (1) smooth affine commutative schemes with geometrically connected fibers, (2) finite flat group schemes, and (3) extensions of abelian schemes by tori, we give explicit descriptions of the second higher direct images. If the rank of the log structure at any geometric point of the base is at most one, we show that the second higher direct image is zero for group schemes in case (1), case (3), and a certain subcase of case (2). If the underlying scheme of the base is of zero-characteristic or of positive characteristic, we can also give more explicit description of the second higher direct image of group schemes in case (1), case (3), and a certain subcase of case (2). Over standard Henselian log traits with finite residue field, we compute the first and the second Kummer log flat cohomology groups with coefficients in group schemes in case (1), case (3), and a certain subcase of case (2).
The automorphism group A of a Fuchsian group Γ of signature (g;m1,...,mr) lifts to a subgroup A˜<Aut(F), where F is the free group of rank 2g+r (Zieshang 1966). We focus on deriving a small finite set of generators of A˜. In the literature, an explicit finite generating set for A˜ was determined in two important subcases: when g=0 (Γ has planar signature), or when r=0 (Γ is a surface group). In this paper, we identify a finite set of generators for g=1 and r≥0. The result allows us to determine both a finite generating set of the mapping class group M(r,1) and that of Aut(Γ), where Γ is a Fuchsian group of a toroidal signature. The finite generating set can be used for computer-aided process to determine equivalence classes of finite groups of symmetries of Riemann surfaces of bounded genus. Some general statements on the stabilisers of certain words in a free group F may find further applications. We establish the result using methods of combinatorial group theory.
