Mixed identities, hereditarily separated actions and oscillation

Abstract Given a topological 𝐺-space, we consider equations with constants over 𝐺. In particular, we formulate some very general conditions on words with constants <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>w</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi>y</m:mi> <m:mo>̄</m:mo> </m:mover> <m:mo>,</m:mo> <m:mover accent="true"> <m:mi>g</m:mi> <m:mo>̄</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> w(\bar{y},\bar{g}) over 𝐺 which guarantee that the inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>w</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mover accent="true"> <m:mi>y</m:mi> <m:mo>̄</m:mo> </m:mover> <m:mo>,</m:mo> <m:mover accent="true"> <m:mi>g</m:mi> <m:mo>̄</m:mo> </m:mover> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≠</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> w(\bar{y},\bar{g})\neq 1 has a solution in 𝐺. These results are illustrated in some typical situations. In particular, standard actions of Thompson’s group 𝐹 and branch groups are considered.