SL_3(Z) contains a non-split sharply 2-transitive subgroup; SL_4(Z) contains a non-split sharply 3-transitive subgroup; SL_3(Z) has no infinite sharply 3-transitive subgroup.
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Non-split sharply 2- and 3-transitive groups in SL_n(\mathbb Z)
SL_3(Z) contains a non-split sharply 2-transitive subgroup; SL_4(Z) contains a non-split sharply 3-transitive subgroup; SL_3(Z) has no infinite sharply 3-transitive subgroup.