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Some model theory of ${\rm SL}(2,{\mathbb R})$

Volume 229 / 2015

Jakub Gismatullin, Davide Penazzi, Anand Pillay Fundamenta Mathematicae 229 (2015), 117-128 MSC: Primary 03C45, 03C64; Secondary 37B99. DOI: 10.4064/fm229-2-2

Abstract

We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We identify a minimal closed $G$-flow $I$ and an idempotent $r\in I$ (with respect to the Ellis semigroup structure $*$ on $S_{G}(M)$). We also show that the “Ellis group” $(r*I,*)$ is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

Authors

  • Jakub GismatullinInstytut Matematyczny
    Uniwersytet Wrocławski
    Pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland
    e-mail
  • Davide PenazziSchool of CEPS
    University of Central Lancashire
    Preston, PR1 2HE, UK
    e-mail
  • Anand PillayDepartment of Mathematics
    University of Notre Dame
    Notre Dame, IN 46556, U.S.A.
    e-mail

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