Abstract Let K be a field of characteristic 0 and let G H connected commutative algebraic groups over . Mor ( , ) denote the set morphisms varieties → that map neutral element to element. We construct natural retraction from Hom( (for arbitrary which commutes with composition addition morphisms. In particular, if are isomorphic as varieties, then they groups. If has no non-trivial unipotent gro...

We introduce a new class of groups with solvable word problem, namely groups specified by a confluent set of short-lex-reducing Knuth– Bendix rules which form a regular language. This simultaneously generalizes short-lex-automatic groups and groups with a finite confluent set of short-lex-reducing rules. We describe a computer program which looks for such a set of rules in an arbitrary finitely...

In this work we introduce a new combinatorial notion of boundary <C of an ω-dimensional cubing C. <C is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of C, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When C arises as the dual of a cubulation – or discrete system of halfspaces – H...

In this paper, we study dense subsets of boundaries of CAT(0) groups. Suppose that a group G acts geometrically on a CAT(0) space X and suppose that there exists an element g0 ∈ G such that (1) Zg0 is finite, (2) X \ Fg0 is not connnected, and (3) each component of X \ Fg0 is convex and not g0-invariant, where Zg0 is the centralizer of g0 and Fg0 is the fixed-point set of g0 in X (that is, Zg0 ...

For n ≥ 0, we exhibit CAT(0) groups that are n-connected at infinity, and have boundary which is (n− 1)-connected, but this boundary has non-trivial n-homotopy group. In particular, we construct 1ended CAT(0) groups that are simply connected at infinity, but have a boundary with non-trivial fundamental group. Our base examples are 1-ended CAT(0) groups that have non-path connected boundaries. I...

Let X be a proper CAT(0) space. A halfspace system (or cubulation) of X is a set H of open halfspaces closed under h 7→ X r h and such that every x ∈ X has a neighbourhood intersecting only finitely many walls of H. Given a cubulation H, one uses the Sageev-Roller construction to form a cubing C(H). One setting in which cubulations were studied is that of a Coxeter group (W,R) acting on its Dav...

A new recursive procedure of the calculation of partition numbers function W (s,dm) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W (s,dm) and their periods τ(G) for the irreducible Coxeter groups and a list for the first ten symmetric group Sm are presented. A least common multiple L(m) of the series of the natural numbers 1,2,.....