Hausdorff dimensions in p-adic analytic groups
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Conversely, it is a long-standing open question whether |hspecP(G)|<∞ implies that G is p-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that G is soluble.
Furthermore, we explore the problem and related questions also for other filtration series, such as the lower p-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for p > 2, that every countably based pro-p group G with an open subgroup mapping onto ℤp ⨁ ℤp admits a filtration series S such that hspecS(G) contains an infinite real interval.
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