Involutary presentations, with applications to Coxeter groups, NEC- groups, and groups of Kanevskiĭ
The dependence problem DP(n) is solvable for a group G if there is an algorithm to decide for any \((k+1)\)-tuple \((g_ 0,g_ 1,...,g_ k)\) of elements of G, with \(k<n\), whether there exist a subsequence \((g_{i_ 1},...,g_{i_ p})\) of \((g_ 1,...,g_ k)\) and elements \(u_ 1,...,u_ p\) of G such that \(g_ 0(u_ 1g_{i_ 1}u_ 1^{- 1})\cdot \cdot \cdot (u_ pg_{i_ p}u_ p^{-1})=1\). DP(1) is the word problem and DP(2) is the conjugacy problem. When DP(n) is solvable for all n, then it is said that DP(\(\infty)\) is solvable for G. The author finds a class of groups for which DP(\(\infty)\) is solvable. These are the groups which have hyperbolic involutary presentations. An involutary presentation associates an inversion involution to the generators, so generators of order \(\leq 2\) are fixed by the involution. This allows certain arguments to be approached in a uniform way, so that generators of order 2 need not be treated as exceptional. An involutary presentation is called hyperbolic when one can attach weights, satisfying certain conditions, to (all cyclic permutations of) the relations. The conditions are stated in terms of the coinitial graph, or star complex, associated to the presentation. They allow simplifications to be made on diagrams representing words which are trivial in G. (An additional technical condition, which is implied if each generator whose square is a relation is nontrivial in G, is needed to deduce DP(\(\infty).)\) It is proved that Coxeter groups of extra large type (the product of two adjacent generating involutions has order at least 4) and most of the ones of large type (the product has order at least 3) admit hyperbolic involutary presentations. So do the cocompact NEC groups, i.e. the finitely generated discrete groups of isometries of the hyperbolic plane with compact orbit space. Thus DP(\(\infty)\) is solvable for these examples. The author also studies certain groups of automorphisms of cubic surfaces, first investigated by \textit{D. S. Kanevskij} [J. Algebra 80, 559-565 (1983; Zbl 0549.14006)]. The involutary presentations of these are not hyperbolic; however, using his techniques the author gives short proofs of some results of Kanevskij including the solvability of the word problem, and that the only torsion elements in these groups are the obvious ones. The author raises the question of whether for \(n\geq 2\) there exist finitely presented groups for which DP(n) is solvable but \(DP(n+1)\) is not.
groups of automorphisms of cubic surfaces
scientific article; zbMATH DE number 4112879