Coarse equivalence and topological couplings of locally compact groups
M. Gromov has shown that any two finitely generated groups $$\Gamma $$Γ and $$\Lambda $$Λ are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions $$\Gamma \curvearrowright X \curvearrowleft \Lambda $$Γ↷X↶Λ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second-countable groups.
GEOMETRIES OF TOPOLOGICAL GROUPS
1. Banach spaces as geometric objects 1 1.1. Categories of geometric structures 1 1.2. Metric spaces 2 1.3. Lipschitz structures 3 1.4. Banach spaces as uniform spaces 5 1.5. Banach spaces as coarse spaces 6 1.6. Rigidity of morphisms and embeddability 8 1.7. Banach spaces as local objects 11 2. Geometric structures on topological groups 12 2.1. Uniform and locally Lipschitz structure 12 2.2. Coarse and quasimetric structure 16 2.3. Lipschitz geometry 21 References 22
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.