C-graph automatic groups

Abstract

Abstract. We generalize the notion of a graph automatic group introduced by Kharlampovich, Khoussainov and Miasnikov by replacing the regular languages in their definition with more powerful language classes. For a fixed language class C, we call the resulting groups C-graph automatic. We prove that the class of C-graph automatic groups is closed under change of generating set, direct and free product for certain classes C. We show that for quasirealtime counter-graph automatic groups where normal forms have length that is linear in the geodesic length, there is an algorithm to compute normal forms (and therefore solve the word problem) in polynomial time. The class of quasirealtime counter-graph automatic groups includes all Baumslag-Solitar groups, and the free group of countably infinite rank. Context-sensitive-graph automatic groups are shown to be a very large class, which encompasses, for example, groups with unsolvable conjugacy problem, the Grigorchuk group, and Thompson's groups F, T and V .

Cayley Automatic Representations of Wreath Products

“…It can be seen that #supp f = p + q. Therefore, we obtain (7). Let us prove the inequality |g| 3|w| − 2.…”

“…In [7], Elder and Taback considered the extensions of the notion of Cayley automatic groups replacing the regular languages by more powerful languages. We denote by C a class of languages; for example, it can be the class of regular languages, context-free languages or context-sensitive languages.…”

“…Notice that for context-free and indexed Cayley automatic groups Definition 1 depends on the choice of generators. In terms of [7], the group G, the subset S and the finite alphabet Σ in Definition 1 form a C-graph automatic triple.…”

Higher rank lamplighter groups are graph automatic

“…Any set of normal forms forming the basis of an automatic structure for a group G is automatically quasigeodesic. The proof of Lemma 8.2 of [13] contains the observation that graph automatic groups naturally possess a quasigeodesic normal form, and a proof is included in [5].…”

“…where α k is computed in Equation (5). As this expression has no terms of negative degree, we do not need to consider b(t+l i ) −1 when computing Q n for n = i, d. When n = d we use Equation (6) in place of Equation (5) and observe that there are no terms of nonpositive degree in LS d (b(t + l i ) −1 ). Hence b(t + l i ) −1 does not play a role in determining Q d .…”