Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

Isomorphism classification of Leary–Minasyan groups

Recently, I. J. Leary and A. Minasyan [Commensurating HNN extensions: Nonpositive curvature and biautomaticity, Geom. Topol. 25 (2021), 4, 1819–1860] studied the class of groups G ⁢ ( A , L ) G(A,L) defined as commensurating HNN-extensions of Z n \mathbb{Z}^{n} . This class, containing the class of Baumslag–Solitar groups, also includes other groups with curious properties, such as being CAT(0) but not biautomatic. In this paper, we classify the groups G ⁢ ( A , L ) G(A,L) up to isomorphism.

Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

On finite state automaton actions of HNN extensions of free abelian groups

HNN extensions of free abelian groups are considered. For arbitrary prime $p$ it is introduced a class of such extensions that act by finite automaton permutations over an alphabet $ \mathsf{X} $ of cardinality $p$ and belong to $p$-Sylow subgroup of the group of automaton permutations over such $ \mathsf{X} $. As a corollary it implies that all corresponding HNN extensions are residually $p$-finite.

Existentially closed Leibniz algebras and an embedding theorem

In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.