The action of matrix groups on aspherical manifolds

Ye, Shengkui

doi:10.2140/agt.2018.18.2875

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The action of matrix groups on aspherical manifolds

Shengkui Ye

Algebraic & Geometric Topology 18 (2018) 2875–2895

DOI: 10.2140/agt.2018.18.2875

Abstract

Let ${SL}_{n} (Z)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mo class="qopname"> SL</mo></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℤ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ for $n \geq 3$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi> <mo class="MathClass-rel">≥</mo> <mn>3</mn></math>$ be the special linear group and $M^{r}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>M</mi></mrow><mrow><mi>r</mi></mrow></msup></math>$ be a closed aspherical manifold. It is proved that when $r < n$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi> <mo class="MathClass-rel">&lt;</mo> <mi>n</mi></math>$ , a group action of ${SL}_{n} (Z)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mo class="qopname"> SL</mo></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℤ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ on $M^{r}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>M</mi></mrow><mrow><mi>r</mi></mrow></msup></math>$ by homeomorphisms is trivial if and only if the induced group homomorphism ${SL}_{n} (Z) \to Out (π_{1} (M))$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mo class="qopname"> SL</mo></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℤ</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">→</mo><mo class="qopname"> Out</mo><mrow><mo class="MathClass-open">(</mo><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>M</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. In particular, when $π_{1} (M)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>M</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ is nilpotent, the group ${SL}_{n} (Z)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mo class="qopname"> SL</mo></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>ℤ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ cannot act nontrivially on $M$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>$ when $r < n$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>r</mi> <mo class="MathClass-rel">&lt;</mo> <mi>n</mi></math>$ . This confirms a conjecture related to Zimmer’s program for these manifolds.

Keywords

Nil-manifolds, aspherical manifolds, Zimmer's program, matrix group actions

Mathematical Subject Classification 2010

Primary: 57S25, 57S20

Secondary: 57S17

References

Publication

Received: 20 June 2017

Revised: 19 April 2018

Accepted: 3 June 2018

Published: 22 August 2018

Authors

Shengkui Ye
Department of Mathematical Sciences Xi’an Jiaotong-Liverpool University Jiangsu China
https://yeshengkui.wordpress.com/

Volume 18, issue 5 (2018)