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Classification of 6-dimensional splittable flat solvmanifolds

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Abstract

A flat solvmanifold is a compact quotient \(\Gamma \backslash G\) where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and \(\Gamma \) is a lattice of G. Any such Lie group can be written as \(G={\mathbb {R}}^k < imes _{\phi } {\mathbb {R}}^m\) with \({\mathbb {R}}^m\) the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice \(\Gamma \) that can be decomposed as \(\Gamma =\Gamma _1 < imes _{\phi }\Gamma _2\), where \(\Gamma _1\) and \(\Gamma _2\) are lattices of \({\mathbb {R}}^k\) and \({\mathbb {R}}^m\), respectively. We analyze the relation between these lattices and the conjugacy classes of finite abelian subgroups of \(\mathsf{GL}(n,{\mathbb {Z}})\), which is known up to \(n\le 6\). From this we obtain the classification of 6-dimensional splittable flat solvmanifolds.

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Notes

  1. An affine equivalence between two Riemannian manifolds M and N is a diffeomorphism \(F:M\rightarrow N\) such that \(f^*\nabla ^M =\nabla ^N\), where \(\nabla ^M\) (respectively \(\nabla ^N\)) is the Levi–Civita connection on M (respectively N).

  2. A Lie algebra \({\mathfrak {g}}\) is said to be unimodular if \({\text {tr}}{\text {ad}}_X=0\) for all \(X\in {\mathfrak {g}}\).

  3. We will denote \(A\oplus B\) the block diagonal matrix \(\begin{pmatrix}A&{}0\\ 0&{}B\end{pmatrix}\).

  4. A \(n\times n\) matrix A will be said to be similar (or conjugated) to B if there exists \(P\in \mathsf{GL}_n({\mathbb {R}})\) such that \(P^{-1}AP=B\) and integrally similar if \(P\in \mathsf{GL}_n({\mathbb {Z}})\).

  5. Given a matrix A, \(P_A\) and \(M_A\) will denote the characteristic and the minimal polynomials of A respectively.

  6. See http://primefan.tripod.com/TotientAnswers1000.html.

  7. We include the computations we made to obtain the classification of the integral similarity classes of integer matrices obtained by conjugating \(\phi (t_0)\) because we believe that these calculations are useful for tackling the same problem in higher dimensions where there are no classifications available.

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Acknowledgements

I am very grateful to my Ph.D. advisor Adrián Andrada for his continuous guidance during the writing of this paper. I also thank Jonas Deré and Derek Holt for the useful conversations and Examples 3.3 and 3.8, respectively.

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  1. FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, X5000HUA, Córdoba, Argentina

    Alejandro Tolcachier

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  1. Alejandro Tolcachier
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Tolcachier, A. Classification of 6-dimensional splittable flat solvmanifolds. manuscripta math. 170, 531–561 (2023). https://doi.org/10.1007/s00229-021-01364-w

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