‣ PosetFpGroup( G ) | ( operation ) |
‣ PosetPresentation( P ) | ( operation ) |
Returns the poset associated to a FpGroup or group presentation. It is defined as the face poset of the barycentric subdivision of the standard complex associated to a finite group presentation. The presentation poset has height 2 and its fundamental group is isomorphic to \(G\). For more details on this construction, see [Fer17, Definition 6.1.1]. For a finitely presented group G, it returns a finite poset of height 2 whose fundamental group is isomorphic to G.
gap> F:=FreeGroup("x");; gap> AssignGeneratorVariables(F);; #I Assigned the global variables [ x ] gap> G:=F/[x^2];; gap> P:=PosetFpGroup(G); <finite poset of size 13> gap> pi1:=FundamentalGroup(P); #I there are 1 generator and 1 relator of total length 2 <fp group on the generators [ f1 ]> gap> Size(pi1); 2
Discrete Morse Theory can be applied to obtain isomorphic presentations to a given finite group presentation. Moreover, under some hypothesis on the matching, the resulting presentation is in the same AC-class of the given one.
‣ MorsePresentation( G, M ) | ( function ) |
Takes an FpGroup G and an acyclic spanning matching with only one critical cell of dimensionvertex of height 0 in the poset PosetFpGroup(G) associated to G. Returns an FpGroup isomorphic to G given by the presentation associated to the \(2\)-complex obtained from \(X\) by collapsing cells according to M. Moreover, the resulting presentation belongs to the same Andrews-Curtis class than the presentation associated to G.
‣ GreedySpanningMatching( X ) | ( function ) |
Tries to find a spanning matching for the poset X.
Colorings can be applied to obtain isomorphic presentations to a given finite group presentation. Moreover, under some hypothesis on the coloring, the resulting presentation is in the same AC-class of the given one.
‣ ColoringPresentation( G, A ) | ( function ) |
Takes an FpGroup G and a simply-connected coloring A in the poset PosetFpGroup(G) associated to G and returns an FpGroup isomorphic to G. Moreover, if A is a spanning collapsible subdiagram of PosetFpGroup(G), then the resulting presentation of FpGroup belongs to the same Andrews Curtis class as the presentation of G.
The reference for this chapter is [Fer17].
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