Endomorphic Presentations of Branch Groups

We introduce “endomorphic presentations,” or L-presentations: group presentations whose relations are iterated under a set of substitutions on the generating set, and show that a broad class of groups acting on rooted trees admit explicitly constructible finite L-presentations, generalising results by Igor Lysionok and Said Sidki.

[1]  R. Lyndon,et al.  Combinatorial Group Theory , 1977 .

[2]  Wilhelm Magnus,et al.  The History of Combinatorial Group Theory , 1982 .

[3]  G. Baumslag.  A finitely generated, infinitely related group with trivial multiplicator , 1971, Bulletin of the Australian Mathematical Society.

[4]  V. Sergiescu.  Graphes planaires et présentations des groupes de tresses , 1993 .

[5]  G. Higman.  Subgroups of finitely presented groups , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  Spectra of non-commutative dynamical systems and graphs related to fractal groups , 2000, math/0012174.

[7]  N. Blackburn.  Some homology groups of wreathe products , 1972 .

[8]  Пример конечно определенной аменабельной группы, не принадлежащей классу $EG$@@@An example of a finitely presented amenable group not belonging to the class $EG$ , 1998 .

[9]  R. I. Grigorchuk,et al.  Just Infinite Branch Groups , 2000 .

[10]  Laurent Bartholdi,et al.  On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups , 1999 .

[11]  W. Magnus,et al.  Varieties of Groups , 1982 .

[12]  Dan Segal,et al.  New Horizons in pro-p Groups , 2000 .

[13]  A Proof of Higman's Embedding Theorem Using Britton Extensions of Groups , 1973 .

[14]  W. Burnside.  Note on the Symmetric Group , 1896 .

[15]  Groups St Andrews 1997 in Bath, I: On the system of defining relations and the Schur multiplier of periodic groups generated by finite automata , 1999 .

[16]  Mark V. Sapir,et al.  Length and Area Functions on Groups and Quasi-Isometric Higman Embeddings , 2001, Int. J. Algebra Comput..

[17]  On the failure of the co-hopf property for subgroups of word-hyperbolic groups , 2001 .

[18]  Gregory Karpilovsky,et al.  The Schur multiplier , 1987 .

[19]  Stäl Aanderaa,et al.  Modular Machines and The Higman-Clapham-Valiev Embedding Theorem , 1980 .

[20]  E. H. Moore,et al.  Concerning the Abstract Groups of Order k ! and ½k ! Holohedrically Isomorphic with the Symmetric and the Alternating Substitution-Groups on k Letters , 1896 .

[21]  A. Lindenmayer,et al.  Cellular Automata, Formal Languages and Developmental Systems , 1973 .

[22]  Leonard Evens,et al.  Cohomology of groups , 1991, Oxford mathematical monographs.

[23]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[24]  Grzegorz Rozenberg,et al.  The mathematical theory of L systems , 1980 .

[25]  Narain Gupta.  Recursively presented two generated infinitep-groups , 1984 .

[26]  Colin Campbell.  Groups St Andrews 1997 in Bath , 1999 .

[27]  Gilbert Baumslag,et al.  Wreath products and finitely presented groups , 1961 .

[28]  S. Sidki.  On a 2-generated infinite 3-group: The presentation problem , 1987 .

[29]  P. Hall,et al.  Finiteness Conditions for Soluble Groups , 1954 .

[30]  S. Sidki,et al.  A just-nonsolvable torsion-free group defined on the binary tree , 1999 .

[31]  R. Grigorchuk.  Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means , 1985 .

[32]  Laurent Bartholdi,et al.  ON THE WORD AND PERIOD GROWTH OF SOME GROUPS OF TREE AUTOMORPHISMS , 2001 .

[33]  I. Lysenok.  A system of defining relations for a Grigorchuk group , 1985 .