On hyperbolic once-punctured-torus bundles IV: Automata for lightning curves

Let 〈A,B,C〉:=〈A,B,C,D|A2=B2=C2=ABCD=1〉. Let R and L denote the automorphisms of 〈A,B,C〉 determined by , . Let (a1,b1,a2,b2,…,ap,bp) be a non-empty, even-length, positive-integer sequence, let F denote Ra1Lb1Ra2Lb2⋯RapLbp, and let 〈A,B,C,F〉 denote the semidirect product . In an influential unfinished work, Jørgensen constructed a discrete faithful representation ρF:〈A,B,C,F〉→PSL2(C). The group 〈A,B,C,F〉 then acts conformally on the Riemann sphere via ρF. Using results of Thurston, Minsky, McMullen, Bowditch, and others, Cannon–Dicks showed that has a CW-structure formed from three closed two-cells, denoted [A], [B] and [C], that are Jordan disks satisfying the ping-pong conditions A[A]=[B]∪[C], B[B]=[C]∪[A], and C[C]=[A]∪[B]. Further, Cannon–Dicks expressed the resulting theta-shaped one-skeleton as the union of two arcs, denoted ∂−A and ∂+B, and expressed each of these lightning curves as limit sets of finitely generated subsemigroups of 〈A,B,C,F〉. The foregoing results had previously been obtained by Alperin–Dicks–Porti for F=RL by elementary methods. Independently, Mumford, Scorza, Series, Wright, and others studied more general lightning curves that arise as limits of sequences of finite chains of round disks in . Later, Cannon–Dicks showed that the set of 〈D,F〉-translates of ∂−A∪∂+B gives a tessellation CW(F) of C with tiles that are Jordan disks.In this article, we find that classic Adler–Weiss automata codify ∂−A and ∂+B in terms of ends of trees. The ∂+B-automaton distinguishes a tree of words in a certain finite alphabet S that is a subset of 〈A,B,C,F〉. The ∂−A-automaton distinguishes a tree of words in the finite alphabet S−1. The automata allow depth-first searches which give drawings of ∂−A and ∂+B that, while requiring less computer time and memory, are more detailed than those that have hitherto been obtained.We show that the limit set of the semigroup generated by S is ∂+B and the limit set of the semigroup generated by S−1 is ∂−A. We use this to show that the Hausdorff dimensions of ∂−A and ∂+B are equal. We raise the problem of whether or not the common Hausdorff dimension can be calculated by applying a famous technique of McMullen to the ∂−A-automaton.We note that the improved drawings of ∂−A and ∂+B give improved drawings of the planar tessellation CW(F). We review Rileyʼs sufficient condition for the columns of CW(F) to be vertical. We review Hellingʼs description of Jørgensenʼs ρRLn and Hodgson–Meyerhoff–Weeksʼ ρRL∞, and we draw CW(RL100) together with something we call CW(RL∞).