Computing subgroup presentations, using the coherence arguments of McCammond and Wise

Abstract We describe an algorithm which may be used to compute a finite presentation of a finitely generated subgroup of a finitely presented group G , provided that G satisfies appropriate hypotheses. The algorithm is based on an algorithm of McCammond and Wise, but is extended to cover a wider class of groups, including all those satisfying the path reduction or weak 2-cell reduction hypotheses of McCammond and Wise. The proofs of correctness of our algorithm emerge from McCammond and Wise' own proofs that their hypotheses imply coherence of the groups satisfying them. We also demonstrate that the algorithm may be extended further to cover groups satisfying appropriate conditions on fans (strings of 2-cells) within disc diagrams. The core of this work originally appeared in the PhD thesis of the first author.

[1]  Neil Ghani,et al.  A Rewriting Alternative to Reidemeister-Schreier , 2003, RTA.

[2]  S. M. Gersten,et al.  Combinatorial group theory and topology , 1987 .

[3]  Leonard H. Soicher,et al.  GRAPE: A System for Computing with Graphs and Groups , 1991, Groups And Computation.

[4]  Derek F. Holt,et al.  Computing Automatic Coset Systems and Subgroup Presentations , 1999, J. Symb. Comput..

[5]  G. P. Scott.  Finitely Generated 3‐Manifold Groups are Finitely Presented , 1973 .

[6]  Leonard H. Soicher,et al.  An Algorithmic Approach to Fundamental Groups and Covers of Combinatorial Cell Complexes , 2000, J. Symb. Comput..

[7]  Pascal Weil,et al.  PSPACE-complete problems for subgroups of free groups and inverse finite automata , 2000, Theor. Comput. Sci..

[8]  S. Margolis,et al.  On the Kurosh theorem and separability properties , 2003 .

[9]  O. Schreier,et al.  Die Untergruppen der freien Gruppen , 1927 .

[10]  J. Stillwell.  Classical topology and combinatorial group theory , 1980 .

[11]  John R. Stallings,et al.  THE TODD-COXETER PROCESS, USING GRAPHS , 1987 .

[12]  Derek F. Holt.  The Warwick automatic groups software , 1994, Geometric and Computational Perspectives on Infinite Groups.

[13]  Charles C. Sims,et al.  Computation with finitely presented groups , 1994, Encyclopedia of mathematics and its applications.

[14]  Daniel T. Wise,et al.  Fans and Ladders in Small Cancellation Theory , 2002 .

[15]  Manuel Delgado,et al.  Combinatorial Group Theory, Inverse Monoids, Automata, And Global Semigroup Theory , 2002, Int. J. Algebra Comput..

[16]  Daniel T. Wise,et al.  Coherence, local quasiconvexity, and the perimeter of 2-complexes , 2002 .

[17]  Mapping tori of free group automorphisms are coherent , 1999, math/9905209.

[18]  John R. Stallings,et al.  Topology of finite graphs , 1983 .

[19]  R. Lathe.  Phd by thesis , 1988, Nature.

[20]  B. Sury.  Combinatorial group theory , 1996 .