Counting maximal arithmetic subgroups

Abstract

We study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semi-simple Lie group using an extension of the method due to

Counting arithmetic lattices and surfaces

“…This result was greatly extended by Borel and Prasad [BP89]. In recent years there has been an attempt to quantify Wang's theorem and to give some estimates on L H .x/ (see [BGLM02], [Gel04], [GLNP04], [Bel07] and [BL]). …”

“…Some yet unproved number-theoretic conjectures imply that a polynomial upper bound is true also in the first case (see [Bel07]). Theorem 1.6 was proved in [Bel07] for higher absolute rank groups but the cases of PSL 2 ‫/ޒ.‬ and PSL 2 ‫/ރ.‬ which are the most crucial for us were left open; in particular, Theorem 1.6 answers a question from [Bel07].…”

“…Theorem 1.6 was proved in [Bel07] for higher absolute rank groups but the cases of PSL 2 ‫/ޒ.‬ and PSL 2 ‫/ރ.‬ which are the most crucial for us were left open; in particular, Theorem 1.6 answers a question from [Bel07].…”

“…The general strategy of the proof of Theorem 1.6 is similar to [Bel07] but some special considerations are needed for small rank. (Note that in [Bel07] the proof is easier for very high rank; see Proposition 3.3 there.)…”

“…(Note that in [Bel07] the proof is easier for very high rank; see Proposition 3.3 there.) The seminal work of Borel [Bor81], which gives a detailed description of the maximal arithmetic lattices in PGL 2 ‫/ޒ.‬ a PGL 2 ‫/ރ.‬ b combined with some ideas of Chinburg and Friedman [CF86], enables us to prove it.…”