An Introduction to Automatic Groups

We extend the range of observed behaviour among length functions of optimal asynchronously automatic structures. We do so by means of a construction that yields asynchronously automatic groups with finite aspherical presentations where the Dehn function of the group is polynomial of arbitrary degree. Many of these groups can be embedded in the automorphism group of a free group. Moreover, the fact that the groups have aspherical presentations makes them useful tools in the search to determine the spectrum of exponents for second order Dehn functions. We contribute to this search by giving the first exact calculations of groups with quadratic and superquadratic exponents. 2000 Mathematical Subject Classification:20F06, 20F65, 20F69.

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Generalized small cancellation presentations for automatic groups

Groups – Complexity – Cryptology ◽

10.1515/gcc-2014-0007 ◽

2014 ◽

Vol 0 (0) ◽

Author(s):

Robert H. Gilman

Keyword(s):

Small Cancellation ◽

Automatic Groups ◽

Finite Presentations

AbstractBy a result of Gersten and Short finite presentations satisfying the usual non-metric small cancellation conditions present biautomatic groups. We show that in the case in which all pieces have length 1, a generalization of the C(3)-T(6) condition yields a larger collection of biautomatic groups.

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Automatic groups and amalgams

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(91)90139-s ◽

1991 ◽

Vol 76 (3) ◽

pp. 229-316 ◽

Cited By ~ 53

Author(s):

G. Baumslag ◽

S.M. Gersten ◽

M. Shapiro ◽

H. Short

Keyword(s):

Automatic Groups

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Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes

International Journal of Algebra and Computation ◽

10.1142/s0218196714500349 ◽

2014 ◽

Vol 24 (06) ◽

pp. 795-813

Author(s):

Yoshiyuki Nakagawa ◽

Makoto Tamura ◽

Yasushi Yamashita

Keyword(s):

Cube Complex ◽

Automatic Group ◽

Mild Conditions ◽

Automatic Groups ◽

Cube Complexes

We discuss a problem posed by Gersten: Is every automatic group which does not contain ℤ × ℤ subgroup, hyperbolic? To study this question, we define the notion of "n-track of length n", which is a structure like ℤ × ℤ, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts freely, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question is answered affirmatively.

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