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Integral Gassman equivalence of algebraic and hyperbolic manifolds. (English) Zbl 1439.58018

Let \(G\) be a group. Given \(R\) a commutative ring, a pair of subgroups \(H_1\) and \(H_2\) of \(G\) is called \(R\)-equivalent if \(R[G/H_1]\) and \(R[G/H_2]\) are isomorphic as \(R[G]\)-modules. When \(G\) is finite, \(H_1,H_2\) are \(\mathbb Q\)-equivalent if and only if \((G,H_1,H_2)\) is a Gassmann triple.
Gassmann triples have been used to construct non-isometric isospectral manifolds (via Sunada’s method), number fields with the same zeta functions, among many other ‘isospectral’ objects. The article under review obtains interesting constructions by using the notion of \(\mathbb Z\)-Gassmann triples. More precisely, the authors construct arbitrarily large families of strongly isospectral and length isospectral hyperbolic \(n\)-manifolds with compatible \(k\)th singular cohomology groups with integers coefficients. They also construct some isospectral objects of algebraic nature.
These results extend previous work by D. Prasad [Adv. Math. 312, 198–208 (2017; Zbl 1430.11153)].

MSC:

58J53 Isospectrality
20G10 Cohomology theory for linear algebraic groups
22E40 Discrete subgroups of Lie groups

Citations:

Zbl 1430.11153
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References:

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