On the isospectra and the isometries of the Aloff-Wallach spaces (Q2732169)

The authors prove the following main theorem: For two Aloff-Wallach spaces \(M_{k,l}\) and \(M_{k',l'}\) the following three statements are equivalent to each other:NEWLINENEWLINENEWLINE(1) \(M_{k,l}\) and \(M_{k',l'}\) are isospectral for the Laplacian acting on smooth functions.NEWLINENEWLINENEWLINE(2) \(M_{k,l}\) is isometric to \(M_{k',l'}\).NEWLINENEWLINENEWLINE(3) \((k',l')\) is one of the pairs \(\pm(k,l)\), \(\pm(k,-(k+l))\), \(\pm(l,-(k+l))\), \(\pm((k+l),k)\), \(\pm((k+l),l)\).NEWLINENEWLINENEWLINEThe nontrivial part is the implication of (1) to (3).