Recent zbMATH articles in MSC 20F28https://www.zbmath.org/atom/cc/20F282021-04-16T16:22:00+00:00WerkzeugStability in outer space.https://www.zbmath.org/1456.200452021-04-16T16:22:00+00:00"Hamenstädt, Ursula"https://www.zbmath.org/authors/?q=ai:hamenstadt.ursula"Hensel, Sebastian"https://www.zbmath.org/authors/?q=ai:hensel.sebastian-wolfgangSummary: We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of \(\mathrm{Out}(F_n)\) as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via their action on Outer space in a way which resembles the characterization of convex cocompact subgroups of mapping class groups.Generators of split extensions of abelian groups by cyclic groups.https://www.zbmath.org/1456.200282021-04-16T16:22:00+00:00"Guyot, Luc"https://www.zbmath.org/authors/?q=ai:guyot.lucSummary: Let \(G\simeq M\rtimes C\) be an \(n\)-generator group which is a split extension of an abelian group \(M\) by a cyclic group \(C\). We study the Nielsen equivalence classes and T-systems of generating \(n\)-tuples of \(G\). The subgroup \(M\) can be turned into a finitely generated faithful module over a suitable quotient \(R\) of the integral group ring of \(C\). When \(C\) is infinite, we show that the Nielsen equivalence classes of the generating \(n\)-tuples of \(G\) correspond bijectively to the orbits of unimodular rows in \(M^{n -1}\) under the action of a subgroup of \(\mathrm{GL}_{n - 1}(R)\). Making no assumption on the cardinality of \(C\), we exhibit a complete invariant of Nielsen equivalence in the case \(M\simeq R\). As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, split metacyclic groups and lamplighter groups.A dense geodesic ray in the \(\mathrm{Out}(F_r)\)-quotient of reduced outer space.https://www.zbmath.org/1456.200332021-04-16T16:22:00+00:00"Algom-Kfir, Yael"https://www.zbmath.org/authors/?q=ai:algom-kfir.yael"Pfaff, Catherine"https://www.zbmath.org/authors/?q=ai:pfaff.catherineSummary: In [Ann. Math. Stud. 97, 417--438 (1981; Zbl 0476.32027)] \textit{H. Masur} proved the existence of a dense geodesic in the moduli space for a surface. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron-Frobenius eigenvectors of positive integer \(m\times m\) matrices is dense in the positive cone \(\mathbf{R}^m_+\) (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the boundary of Outer Space is the limit of a flow line.Algebraic laminations for free products and arational trees.https://www.zbmath.org/1456.200202021-04-16T16:22:00+00:00"Guirardel, Vincent"https://www.zbmath.org/authors/?q=ai:guirardel.vincent"Horbez, Camille"https://www.zbmath.org/authors/?q=ai:horbez.camilleIn analogy to curve complexes used to study mapping class groups of surfaces, the free factor graph of a free group \(F_n\) has recently turned to be fruitful in the study of Out(\(F_n\)). It is Gromov hyperbolic, as was proved by \textit{M. Bestvina} and \textit{M. Feighn} [Adv. Math. 256, 104--155 (2014; Zbl 1348.20028)], and the action of an automorphism of \(F_n\) is loxodromic if and only if it is fully irreducible. Its Gromov boundary was described by \textit{M. Bestvina} and \textit{P. Reynolds} [Duke Math. J. 164, No. 11, 2213--2251 (2015; Zbl 1337.20040)] and \textit{U. Hamenstädt} [``The boundary of the free splitting graph and the free factor graph'', Preprint, \url{arXiv:1211.1630}] as the set of equivalence classes of arational trees.
The main goal of the paper under review is to extend the theory of algebraic laminations to the context of free products. A key point for this intended application says that if two trees have a leaf in common in their dual laminations, and if one of the trees is arational and relatively free, then they are equivariantly homeomorphic.
Reviewer: V. A. Roman'kov (Omsk)On central endomorphisms of a group.https://www.zbmath.org/1456.200262021-04-16T16:22:00+00:00"Russo, Alessio"https://www.zbmath.org/authors/?q=ai:russo.alessioSummary: Let \(\Gamma\) be a normal subgroup of the full automorphism group \(\mathrm{Aut}(G)\) of a group \(G\), and assume that \(\mathrm{Inn}(G)\leq \Gamma\). An endomorphism \(\sigma\) of \(G\) is said to be \(\Gamma\)-central if \(\sigma\) induces the the identity on the factor group \(G/C_G(\Gamma)\). Clearly, if \(\Gamma =\mathrm{Inn}(G)\), then a \(\Gamma\)-central endomorphism is a central endomorphism. In this article the conditions under which a \(\Gamma\)-central endomorphism of a group is an automorphism are investigated.Every group is the outer automorphism group of an HNN-extension of a fixed triangle group.https://www.zbmath.org/1456.200342021-04-16T16:22:00+00:00"Logan, Alan D."https://www.zbmath.org/authors/?q=ai:logan.alan-dA remarkable result of \textit{I. Bumagin} and \textit{D. T. Wise} [J. Pure Appl. Algebra 200, No. 1--2, 137--147 (2005; Zbl 1082.20021)] says that every countable group \(Q\) can be realized as the outer automorphism group of a finitely generated group \(G_Q\). These groups \(G_Q\) have, for a given \(Q\), often special additional properties. The main result of the paper is as follows. Fix a triangle group \(T_i=\langle a,b\mid a^i=b^i=(ab)^i=1\rangle\) with \(i\geq 6\). For every countable group, given by a countable presentation \(P\), there exists an automorphism-induced HNN-extension \(T_P\) of \(T_i\) such that Out\((T_P)\cong Q=\pi_1(P)\) and Aut\((T_P)\cong T_P\rtimes Q\).
The HNN-extensions \(T_P\) of \(T_i\) are explicitly constructed. Interesting enough, the single steps lead to nice residual and malnormal properties of the constructed groups.
Reviewer: Gerhard Rosenberger (Hamburg)