Konstantin Kozlov

On (proper) compactifications of topological groups.

‎‎‎Ellis’s functional approach provides a way to construct compactificats of a topological group G via its \tau_p-representation in a compact space X, where the topology of G coincides with the topology of pointwise convergence induced by the action G\curvearrowright X. In this setting, elements of G are viewed as points of X^X endowed with Tychonoff topology. The reulting compactifications are right-topological semigroup compactifications, commonly referred to as Ellis compactifications.

‎‎‎An alternative method for constructing compactifications of a topological group G is based on its \tau_{co}-representation in a compact space X, where the topology of G coincides with the compact-open topology induced by the action G\curvearrowright X. In this approach, elements of G are identified with the graphs of the corresponding homeomorphisms and considered as points in the hyperspace 2^{X\times X} endowed with the Vietoris topology. For every compactification of this type, called a graph compactification, the left and right translations as well as the inversion operation extend continuously. Moreover, any graph compactification of G is dominated by the Roelcke compactification.

‎‎‎Both the Ellis and graph approaches to constructing (proper) compactifications of a topological group G are useful for studying extensions of algebraic operations from G to its compactifications. They also provide descriptions of the Roelcke and WAP compactifications. Furthermore, using dichotomy theorems of A.,V. Arhangel’skii, these descriptions can be effectively applied to investigate topological properties of the corresponding remainders.

As illustrations, we consider subgroups of the permutation group and the automorphism group of a linearly ordered topological space (LOTS) endowed with the topology of pointwise convergence.