Geodesic
BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO-$p$ GROUPS
Forum of Mathematics, Pi, Tome 8 (2020)

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In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro-$p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between $n^{(\log n)^{2}}$ and $e^{n}$. Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro-$p$ branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type $n^{\log n}$. 
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     author = {YIFTACH BARNEA and JAN-CHRISTOPH SCHLAGE-PUCHTA},
     title = {BRANCH {GROUPS,} {ORBIT} {GROWTH,} {AND} {SUBGROUP} {GROWTH} {TYPES} {FOR} {PRO-}$p$ {GROUPS}},
     journal = {Forum of Mathematics, Pi},
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YIFTACH BARNEA; JAN-CHRISTOPH SCHLAGE-PUCHTA. BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO-$p$ GROUPS. Forum of Mathematics, Pi, Tome 8 (2020). doi: 10.1017/fmp.2020.8

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