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. b a Generators: Σ = { a , a − 1 , b , b − 1 } Normal forms: R = { a i b j } ⊂ Σ ∗ In general normal forms for an automatic group G can be any regular language projecting onto G . R ⊂ Σ ∗ → G Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 2 / 11
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. c R = ( ab + b ) c ∗ b b b B C S S → aA | bB A → bB a b B → cB | b S . If input a , goto A ; if b , goto B . A A . If input b , goto B . B . If input b halt; if c , goto B . Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 3 / 11
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 Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 4 / 11
Nil and Sol Fundamental groups corresponding to nil and sol geometries are not automatic, e.g., the Heisenberg group 1 i j | i , j , k ∈ Z H = 0 1 k 0 0 1 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] Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 5 / 11
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 = x 2 � 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 ? Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 6 / 11
A Logical View of the Fellow Traveler Property R ⊂ Σ ∗ → G Σ = { a , a − 1 , b , b − 1 , . . . } w → w The synchronous fellow traveler property is equivalent to. For all a ∈ Σ, R a = { ( u , v ) | ua = v } is accepted by a synchronous 1 2-tape automaton. Same for R = . 2 The first order theory of synchronous m -ary relations is decidable, and first order formulas define synchronous relations. For example the short lex order 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. Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 7 / 11
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. Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 8 / 11
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 R a etc. are synchronous. The Heisenberg group is graph automatic 1 0 1 # 1 5 3 1 1 # # → 0 1 6 0 1 0 1 0 0 1 Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 9 / 11
Some Properties of Graph Automatic Groups Independent of choice of generators 1 The word problem is decidable in quadratic time. 2 Closed under direct and free products, and finite extensions. 3 G ≀ Z , G finite, is graph automatic. 4 Finitely generated class 2 nilpotent groups are graph automatic. 5 Baumslag-Solitar groups B (1 , n ) are graph automatic. 6 Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 10 / 11
Directions (Yi Di Zhang) Generalize the definition of FA-presentable groups by 1 allowing the multiplication table to be context free. This generalization includes all word hyperbolic groups and some infinitely generated groups. Generalize graph automatic groups. For example SL ( n , Z ) fails to be 2 graph automatic because the set of invertible matrices is apparently not a suitable image of a regular language. Enlarge the Cayley graph of SL ( n , Z ) to the graph of all matrices. Robert Gilman (Stevens) Automatic Groups Redux GAGTA 2013 11 / 11
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