Compact domination for groups definable in linear o-minimal structures
Pillay's Conjecture says that every definably compact group \(G\) of dimension \(n\) in a saturated o-minimal expansion \(M\) of an ordered group has a smallest type-definable subgroup \(G^{00}\) of bounded index, and \(G/G^{00}\), when equipped with the logic topology, is a compact Lie group of dimension \(n\). This was shown to be true in several cases, most notably when \(M\) expands an ordered field. Compact domination is a notion arising in this setting to formalize the intuition that the canonical homomorphism \(\pi\) of \(G\) onto \(G/G^{00}\) behaves as a standard part map, making \(G^{00}\) an \textsl{``infinitesimal''} subgroup. The Compact Domination Conjecture just says that \(G\) is compactly dominated by \(G/G^{00}\), meaning that, for every definable subset \(X\) of \(G\), for all points \(y \in G/G^{00}\) outside a set of measure 0, \(\pi^{-1} (y)\) is either contained in \(X\) or in its complement (one refers here to the the unique normalized Haar measure \(m\) on the compact Lie group \(G/G^{00}\)). The paper under review proves this conjecture when \(M\) is a \textsl{linear} o-minimal expansion of an ordered group (that is, for every definable function \(f\) from \(A \subseteq M^h\) to \(M\), there is a partition of \(A\) into finitely many pieces \(A_i\) such that for each \(i\), if \(x\), \(y\), \(x+t\) and \(y+t\) are in the same \(A_i\), \(f(x+t) - f(x) = f(y+t) - f(y)\)). The proof introduces a kind of standard part map \(st_G\) from \(G\) to the real \(n\)-torus \((S^1)^n\), \(n = \dim(G)\), resembling \(\pi\) in the following sense: \(G^{00}\) is the kernel of \(st_G\) and, for all \(A \subseteq (S^1)^n\), \(A\) is closed if and only if its preimage in \(G\) via \(st_G\) is type-definable. The author shows that \(st_G\), and consequently \(\pi\), satisfy the following criterion (sufficient to ensure the compact domination thesis): for every definable \(X \subseteq G\) with \(\dim(X)<n\), the measure \(m\) of the image of \(X\) is 0.
scientific article; zbMATH DE number 5620097