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$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Assume that we have two subgroups $G_1,G_2$ of $\Sp(4,\mathbb{Z})$ that are conjugate in $\GL(4,\mathbb{Q})$ $\big($or in $\SL(4,\mathbb{Z})$ if that helps$\big)$.

Does it follow that their indices are equal: $[\Sp(4,\mathbb{Z}):G_1]=[\Sp(4,\mathbb{Z}):G_2]$? If not, is it at least true that if one index is finite, then so is the other?

Thanks!

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    $\begingroup$ Yes. If both indices are both finite, this is a consequence of Harish-Chandra (which ensures these are lattices) and Mostow's theorem (which implies they are conjugate in the real group, so have the same covolume). To exclude the case one index is finite and the other is infinite, one can invoke Margulis' superrigidity. $\endgroup$
    – YCor
    Commented Jan 9, 2022 at 21:43
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    $\begingroup$ @YCor, that seems to be an answer (I wouldn't have guessed that such high-powered techniques were required, but certainly no more elementary argument occurs to me); would you post it as such? $\endgroup$
    – LSpice
    Commented Jan 12, 2022 at 16:41

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