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This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete…

We show that for every quasivariety K of structures (where both functions and relations are allowed) there is a semilattice S with operators such that the lattice of quasi-equational theories of K (the dual of the lattice of…

We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting…

During the study of Coxeter's friezes, M. Cuntz defined the concept of $\lambda$-quiddities and gave the problem of studying them over some subsets of $\mathbb{C}$. The objective of this text is to carry out this study in the case of some…

A planar semimodular lattice $K$ is \emph{slim} if $\mathsf{M}_3$ is not a sublattice of~$K$. In a recent paper, G. Cz\'edli found four new properties of congruence lattices of slim, planar, semimodular lattices, including the \emph{No…

Functions between groups with the property that all function con- jugates are inverse preserving are called sandwich morphisms. These maps pre- serve a structure within the group known as the sandwich structure. Sandwich structures are left…

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group $G$ admitting a general type action on a hyperbolic space $S$, we show that the induced action of the Frattini subgroup $\Phi(G)$ on $S$…

Let $A$, $B$ be separable C*-algebras, $B$ stable. Elements of the E-theory group $E(A,B)$ are represented by asymptotic homomorphisms from the second suspension of $A$ to $B$. Our aim is to represent these elements by (families of) maps…

We show that a fundamental sandwich algebra has an analogue of a root system of a semisimple Lie algebra. This leads to an analogue of a Weyl group, which we study in another paper.

This is a companion to a recent investigation of K-theoretical invariants for symmetric spaces. We introduce a new class of cycles in K-groups, which are connected to elements of an underlying root lattice. This will be needed for a…

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the…

The set of supercharacter theories of a fixed group $G$ forms a natural lattice. An open question in the study of supercharacter theories is to classify this lattice, and to date, this has only been done for the cyclic groups…

For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility…

Let $\Ga$ be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let $\theints$ be the ring of $S$-integers of~$K$. We define a certain…

We characterize the maximal discrete subgroups of $SO^+(2,n+2)$, which contain the discriminant kernel of an even lattice, which contains two hyperbolic planes over $\mathbb{Z}$. They coincide with the normalizers in $SO^+(2,n+2)$ and are…

In 1995 Grillet introduced the concept of a stratified semigroup as a kind of generalisation of finite nilsemigroups. We extend these ideas here by allowing a more general Base and describe them in terms of extensions of semigroups by…

We will give a definition of quadratic forms on bimodules and prove the sandwich classification theorem for subgroups of the general linear group $\mathrm{GL}(P)$ normalized by the elementary unitary group $\mathrm{EU}(P)$ if $P$ is a…

If $\Gamma$ is any nonuniform lattice in the group ${\rm PU}(2,1)$, let $\overline{\Gamma}$ be the quotient of $\Gamma$ obtained by filling the cusps of $\Gamma$ (i.e. killing the center of parabolic subgroups). Assuming that such a lattice…

We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr S$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are non-discrete. Given…

Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $\Gamma <G$ and $\Lambda < G'$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $\Gamma$ into…