Groups With Indexed Co-Word Problem

Holt, Derek F.; Röver, Claas E.

doi:10.1142/s0218196706003359

Cited by 22 publications

(24 citation statements)

References 16 publications

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“…It also includes the co-indexed and co-context free groups as described in [19,20,27]. These groups have co-word problems accepted by non-deterministic pushdown or nested-stack automata, which can be simulated by deterministic linear bounded automata since as described in these articles, the non-determinism is confined to an initial guessing step.…”

Section: It Follows That Dcs-biautomatic Does Not Imply Solvable Conjmentioning

confidence: 99%

C-graph automatic groups

Elder

Taback

2014

Journal of Algebra

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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 .

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Section: It Follows That Dcs-biautomatic Does Not Imply Solvable Conjmentioning

confidence: 99%

C-graph automatic groups

Elder

Taback

2014

Journal of Algebra

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“…A further paper [6] studies groups whose co-word problem is an indexed language (accepted by a nested stack automaton). In particular it is proved in that paper that for all currently known pairs of co CF-groups G and H, the free product G * H has indexed co-word problem.…”

Section: Introductionmentioning

confidence: 99%

Groups With Context-Free Co-Word Problem

Holt

Rees

Röver

et al. 2005

J. London Math. Soc.

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The class of co-context-free groups is studied. A co-context-free group is defined as one whose coword problem (the complement of its word problem) is context-free. This class is larger than the subclass of context-free groups, being closed under the taking of finite direct products, restricted standard wreath products with context-free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co-context-free groups are known. It is proved that the only examples amongst polycyclic groups or the Baumslag-Solitar groups are virtually abelian. This is done by proving that languages with certain purely arithmetical properties cannot be context-free; this result may be of independent interest.

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“…It cannot be deterministic context-free because deterministic context-free languages are closed under complementation, [17]. Thus, Z × Z is co-context-free in the sense of [16]. The class of co-context-free groups is very interesting in its own, for example it includes the Higman-Thompson group [22].…”

Section: Finitely Generated Virtually Free Groups Are Context-freementioning

confidence: 99%

Context-Free Groups and Their Structure Trees

Diekert

Weiß

2013

Int. J. Algebra Comput.

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Let Γ be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structure tree of optimally nested cuts of Γ. Once the tree is constructed, Bass-Serre theory yields that G is virtually free. This approach simplifies the existing proofs for the fundamental result of Muller and Schupp that characterizes context-free groups as f.g. virtually free groups. Our construction avoids the explicit use of Stallings' structure theorem and it is self-contained.We also give a simplified proof for an important consequence of the structure tree theory by Dicks and Dunwoody which has been stated by Thomassen and Woess. It says that a f.g. group is accessible if and only if its Cayley graph is accessible.

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Groups With Indexed Co-Word Problem

Cited by 22 publications

References 16 publications

C-graph automatic groups

C-graph automatic groups

Groups With Context-Free Co-Word Problem

Context-Free Groups and Their Structure Trees

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