On identities in connected topological groups
Evgenii Reznichenko, Il'ya Zyabrev
TL;DR
The paper resolves a long-standing question about local identities in connected topological groups by constructing a counterexample for odd $n>10^{10}$: there exists a connected topological group $G$ and a neighborhood $V$ of the identity such that $x^n=1$ for all $x$ in $V$, yet $x^n\neq1$ for some $x\in G$. The approach combines a Graev-type invariant metric on free groups, a transfinite closure framework for neighborhoods, and discrete group techniques via free Burnside groups and central extensions to produce a connected quotient with a nonlocal obstruction. Key innovations include the invariant norm $N$ and metric $\rho$, contraction-preserving endomorphisms, and a multistage reduction from free groups to carefully constructed quotient groups that retain a local identity while breaking it globally. The results provide a negative answer to Platonov's generalized Mytselsky problem and expose fundamental limits on local-to-global transfer of identities in topological groups, with implications for the structure theory of connected groups and identity propagation. The methods offer a blueprint for constructing similar local-global counterexamples using Graev metrics and transfinite closures.
Abstract
In 1957, Nemytskii proved the following fact: if in a locally compact or in an Abelian connected group there is a neighborhood of the identity in which some identity holds, then it holds in the entire group. The following question was also posed there: Let G be a connected topological group. In some neighborhood of the identity of the group G the identity $x^3=1$ holds. Is it true that then the identity $x^3=1$ holds in the entire group $G$? The same question is posed for the identity $gx^2 = x^2g$, where $g$ is a fixed element of the group. Platonov formulated the following generalized formulation of the Mytselsky problem: for a topological connected group, is it true that if the identity holds in a neighborhood of the identity, then the identity holds everywhere? In this paper, a negative answer to Platonov's question is given, the following theorem is proven: if $n > 10^{10}$ is odd, then there exists a connected topological group in which the identity $x^n=1$ holds in some neighborhood of unity, but not in the entire group.