A Note on Amalgamated Rings Along an Ideal

Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to non-commutative setting and also give new properties of amalgamated rings.

Monoids of O-type, subword reversing, and ordered groups

We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering, and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products.

Thin monodromy in Sp(4)

We show that some hypergeometric monodromy groups in ${\rm Sp}(4,\mathbf{Z})$ split as free or amalgamated products and hence by cohomological considerations give examples of Zariski dense, non-arithmetic monodromy groups of real rank $2$. In particular, we show that the monodromy group of the natural quotient of the Dwork family of quintic threefolds in $\mathbf{P}^{4}$ splits as $\mathbf{Z}\ast \mathbf{Z}/5\mathbf{Z}$. As a consequence, for a smooth quintic threefold $X$ we show that the group of autoequivalences $D^{b}(X)$ generated by the spherical twist along ${\mathcal{O}}_{X}$ and by tensoring with ${\mathcal{O}}_{X}(1)$ is an Artin group of dihedral type.

Actions of discrete groups on stationary Lorentz manifolds

We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (respectively, lightlike) manifold.

AN EXAMPLE OF A SUBGROUP SEPARABLE BUT NOT CONJUGACY SEPARABLE GROUP

We give an example of a conjugacy separable, but not subgroup separable group. It is an adaptation of an example of Tavgen and the third author from [Closed orbits and finite approximability with respect to conjugacy of free amalgamated products, Math. Notes58 (1995) 1042–1048] to a theorem of Raptis [On finiteness conditions of certain graphs of groups, Internat. J. Algebra Comput.5 (1995) 719–724].