Abstract :
Let 〈 A , B , C 〉 : = 〈 A , B , C , D | A 2 = B 2 = C 2 = A B C D = 1 〉 . Let R and L denote the automorphisms of 〈 A , B , C 〉 determined by ( A , B , C ) R = ( A , B C B , B ) , ( A , B , C ) L = ( B , B A B , C ) . Let ( a 1 , b 1 , a 2 , b 2 , … , a p , b p ) be a non-empty, even-length, positive-integer sequence, let F denote R a 1 L b 1 R a 2 L b 2 ⋯ R a p L b p , and let 〈 A , B , C , F 〉 denote the semidirect product 〈 F | 〉 ⋉ 〈 A , B , C 〉 . In an influential unfinished work, Jørgensen constructed a discrete faithful representation ρ F : 〈 A , B , C , F 〉 → PSL 2 ( C ) . The group 〈 A , B , C , F 〉 then acts conformally on the Riemann sphere C ˆ via ρ F . Using results of Thurston, Minsky, McMullen, Bowditch, and others, Cannon–Dicks showed that C ˆ 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 C ˆ . 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.
s 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.
w 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.
e 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 ρ RL n and Hodgson–Meyerhoff–Weeksʼ ρ RL ∞ , and we draw CW ( RL 100 ) together with something we call CW ( RL ∞ ) .
