Automatic Groups Redux Robert Gilman Stevens Institute of - - PowerPoint PPT Presentation

Automatic Groups Redux

Robert Gilman

Stevens Institute of Technology

Geometric and Asymptotic Group Theory with Applications City College of New York May 28, 2013

Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 1 / 11

Automatic Groups

Normal forms for an automatic group G satisfy the synchronous fellow-traveler property when viewed as paths in its Cayley diagram. a b Generators: Σ = {a, a−1, b, b−1} Normal forms: R = {aibj} ⊂ Σ∗ In general normal forms for an automatic group G can be any regular language projecting onto G. R ⊂ Σ∗ → G

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Regular languages

Regular languages are the closure of finite subsets of Σ∗ under union, product, and generation of submonoid. Regular languages are accepted by finite automata, described by regular expressions, generated by regular grammars, and recognized by a simple programming language. a b b b c S A C B R = (ab + b)c∗b S → aA | bB A → bB B → cB | b

• S. If input a, goto A; if b, goto B.
• A. If input b, goto B.
• B. If input b halt; if c, goto B.

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Formal Languages

A formal language is a subset of the free monoid Σ∗ of all words over a finite alphabet Σ A hierarchy of language classes: Regular One-counter, linear, deterministic context free Context free Indexed, growing context sensitive Context sensitive Recursively enumerable

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Nil and Sol

Fundamental groups corresponding to nil and sol geometries are not automatic, e.g., the Heisenberg group H =      1 i j 1 k 1   | i, j, k ∈ Z    Remedies were proposed by [Bridson and Gilman, 1996] and [Baumslag, Short, Shapiro, 1999]. BBS defined the class of poly-pushdown languages. These have recently been considered by [Ceccherini-Silberstein, Coornaert, Fiorenzi and Schupp] as word problems of finitely generated groups. See also [Brough, 2011]

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Open Problems about Automatic Groups

• 1. Is the conjugacy problem solvable for automatic groups?
• 2. Given a finite presentation of an automatic group, can one decide if this

group is hyperbolic?

• 3. Are all automatic groups biautomatic?
• 4. Is every biautomatic group which does not contain any Z × Z

subgroups, hyperbolic?

• 5. Can the group x, y | yxy −1 = x2 be a subgroup of an automatic

group?

• 6. Is a retract of an automatic group automatic?
• 7. Is every finitely presented metabelian automatic group virtually abelian ?

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A Logical View of the Fellow Traveler Property

Σ = {a, a−1, b, b−1, . . .} R ⊂ Σ∗ → G w → w The synchronous fellow traveler property is equivalent to.

1

For all a ∈ Σ, Ra = {(u, v) | ua = v} is accepted by a synchronous 2-tape automaton.

2

Same for R=. The first order theory of synchronous m-ary relations is decidable, and first

• rder formulas define synchronous relations. For example the short lex
• rder on Σ∗, L(x, y), is a synchronous binary relation, so

φ(x) = ∀y R=(x, y) → L(x, y) defines an automatic structure with uniqueness. Unfortunately the language is not very expressive. E.g., we cannot express the binary relation of conjugacy.

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FA-presentable groups

[Khoussainov, Nerode, 1994] Require that the relations I(x, y) = {(x, y) | xy = 1} and M(x, y, z) = {(x, y, z) | xyz = 1 are synchronous. Now conjugacy can be defined. But finitely generated groups of this type are virtually abelian. [Oliver and Thomas, 2005], [Nies and Thomas, 2008] On the other hand any map R → G is allowable.

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Graph Automatic Groups

[Kharlampovich, Khoussainov and Miasnikov, 2011] [Miasnikov and ˇ Suni´ c, 2011] Keep the original definition of automatic group, but allow any map R → G for which the relations Ra etc. are synchronous. The Heisenberg group is graph automatic 1 1 # 1 1 # # 1 1 →   1 5 3 1 6 1  

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Some Properties of Graph Automatic Groups

1

Independent of choice of generators

2

The word problem is decidable in quadratic time.

3

Closed under direct and free products, and finite extensions.

4

G ≀ Z, G finite, is graph automatic.

5

Finitely generated class 2 nilpotent groups are graph automatic.

6

Baumslag-Solitar groups B(1, n) are graph automatic.

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Directions

1

(Yi Di Zhang) Generalize the definition of FA-presentable groups by allowing the multiplication table to be context free. This generalization includes all word hyperbolic groups and some infinitely generated groups.

2

Generalize graph automatic groups. For example SL(n, Z) fails to be graph automatic because the set of invertible matrices is apparently not a suitable image of a regular language. Enlarge the Cayley graph

• f SL(n, Z) to the graph of all matrices.

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