Regular path systems and (bi)automatic groups.

*(English)*Zbl 1165.20036From the introduction: Few systematic methods are known to derive the fact that a group acting on a space is automatic (or biautomatic) out of the properties of the space. One of them is the method of automatic (or biautomatic) fundamental groupoids. However, this method is limited to the case of free actions only. To deal with actions that allow nontrivial, though finite, stabilizers one would like to have some orbispace analogue of this method, which seems to be not yet developed. Apart from applications to new classes of spaces, such a method could be potentially useful in classical cases as well. For example, it is still unknown whether a finite extension of a biautomatic group is biautomatic. Thus,
even knowing that groups acting freely on some classes of spaces are biautomatic, it is in general not possible to claim the same for actions with nontrivial stabilizers (even when the existence of finite index subgroups acting freely is easy to establish).

In this paper we present a method as desired above. It is based on the newly introduced notions of automata over graphs and regular path systems in graphs and other spaces. The main tool of this method, Theorem 1.1, allows us to conclude that a group acting on a space properly discontinuously and cocompactly is automatic (or biautomatic) if there is an appropriate regular path system in the space. The details and proofs related to the method are presented in Sections 1-3.

As an addition to the above method we present also a (less inventive) idea of local recognition for paths in a space, proving that locally recognized path systems are regular (Section 7). This idea allows to establish automaticity (or biautomaticity) of some groups in purely geometric terms, as formulated in Corollary 7.2.

Using the method of this paper we prove new results in several directions. First, we extend some previously known results concerning biautomaticity to bigger classes of groups, by relaxing assumption that a group acts freely to assumption that it acts properly discontinuously (Subsections 8.1 and 8.3). Second, we show that groups acting properly discontinuously and cocompactly on so-called 6-systolic simplicial complexes are biautomatic (Subsection 8.2). Third, we prove that type-preserving groups acting in some natural way on arbitrary buildings are automatic, and discuss a sufficient condition for them to be biautomatic (Section 6). Finally, we give a sufficient condition for a finite semidirect extension of a biautomatic group to be biautomatic (Section 4, Corollary 4.5), and use it to prove that the wreath product of a biautomatic group by a finite group is biautomatic (Section 5). We also prove that if quotient of a group by a finite normal subgroup is biautomatic then the group is also biautomatic (Section 4, Corollary 4.3).

As a curiosity, we present (in Subsection 8.4) an alternative proof of the well-known fact that any Gromov hyperbolic group is biautomatic, based on the ideas developed in the paper.

In this paper we present a method as desired above. It is based on the newly introduced notions of automata over graphs and regular path systems in graphs and other spaces. The main tool of this method, Theorem 1.1, allows us to conclude that a group acting on a space properly discontinuously and cocompactly is automatic (or biautomatic) if there is an appropriate regular path system in the space. The details and proofs related to the method are presented in Sections 1-3.

As an addition to the above method we present also a (less inventive) idea of local recognition for paths in a space, proving that locally recognized path systems are regular (Section 7). This idea allows to establish automaticity (or biautomaticity) of some groups in purely geometric terms, as formulated in Corollary 7.2.

Using the method of this paper we prove new results in several directions. First, we extend some previously known results concerning biautomaticity to bigger classes of groups, by relaxing assumption that a group acts freely to assumption that it acts properly discontinuously (Subsections 8.1 and 8.3). Second, we show that groups acting properly discontinuously and cocompactly on so-called 6-systolic simplicial complexes are biautomatic (Subsection 8.2). Third, we prove that type-preserving groups acting in some natural way on arbitrary buildings are automatic, and discuss a sufficient condition for them to be biautomatic (Section 6). Finally, we give a sufficient condition for a finite semidirect extension of a biautomatic group to be biautomatic (Section 4, Corollary 4.5), and use it to prove that the wreath product of a biautomatic group by a finite group is biautomatic (Section 5). We also prove that if quotient of a group by a finite normal subgroup is biautomatic then the group is also biautomatic (Section 4, Corollary 4.3).

As a curiosity, we present (in Subsection 8.4) an alternative proof of the well-known fact that any Gromov hyperbolic group is biautomatic, based on the ideas developed in the paper.

##### MSC:

20F65 | Geometric group theory |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

57M07 | Topological methods in group theory |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

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