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.
Cohomological induction and uniform measure equivalence
We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.