Lipschitz structure and minimal metrics on topological groups (Q1749372)

A metrisable topological group \(G\) always admits a compatible left-invariant metric \(d\), i.e., the topology of \(G\) is induced by \(d\) and \(d(hg,hf)=d(g,f)\) for all \(g,f,h\in G\). For any two left-invariant metrics \(d\) and \(\partial\) on \(G\) the identity map id\({}:(G,\partial)\to(G,d)\) is a uniform homomorphism. However, in general there is no control of the modulus of the uniform continuity of the mapping. A compatible left-invariant metric \(d\) on a topological group \(G\) is said to be minimal, if for every other compatible left-invariant metric \(\partial\) the identity map id\({}:(G,\partial)\to(G,d)\) is Lipschitz in a neighbourhood of the identity \(1\) of \(G\), i.e., there are a neighbourhood \(U\) of \(1\) and \(K>0\) so that \(d(g,f)\leq K\cdot\partial(g,f)\) for all \(g,f\in U\); \(d\) is said to be maximal, if for every other compatible left-invariant metric \(\partial\) on \(G\) the map id\({}:(G,\partial)\to(G,d)\) is Lipschitz for large distances, i.e., \(d\leq K\cdot\partial+C\) for some constants \(K\) and \(C\). The author arrives to these notions by analyzing the question whether a metrisable group has a canonically defined Lipschitz geometry. He characterizes minimal metrics as those satisfying a certain linear growth condition on powers in a neighbourhood of \(1\). This condition has already been studied in the literature in the context of locally compact groups and is connected to the solution to Hilbert's fifth problem. Combining this with work on the large scale geometry of topological groups the author identifies the class of metrisable groups admitting a canonical global Lipschitz geometry.