Undecidability everywhere F.p. groups Words Word problem Markov - PowerPoint PPT Presentation

Undecidability everywhere Bjorn Poonen Wang tiles Group theory Undecidability everywhere F.p. groups Words Word problem Markov properties Topology Homeomorphism Bjorn Poonen problem Knot theory Algebraic geometry University of California at Berkeley Varieties Isomorphism problem Commutative Cantrell Lecture 3 algebra F.g. algebras University of Georgia F.g. fields March 28, 2008 References

Undecidability Wang tiles everywhere Bjorn Poonen Wang tiles Can you tile the entire plane with copies of the following? Group theory F.p. groups Words Word problem Markov properties Topology Homeomorphism problem Knot theory Algebraic geometry Varieties Isomorphism problem Commutative algebra F.g. algebras Rules: F.g. fields References Tiles may not be rotated or reflected. Two tiles may share an edge only if the colors match.

Undecidability everywhere Bjorn Poonen Conjecture (Wang 1961) Wang tiles If a finite set of tiles can tile the plane, there exists a Group theory periodic tiling. F.p. groups Words Word problem Assuming this, Wang gave an algorithm for deciding whether Markov properties a finite set of tiles can tile the plane. Topology Homeomorphism problem Knot theory Algebraic geometry But. . . Varieties Isomorphism problem Commutative algebra F.g. algebras F.g. fields References

Undecidability everywhere Bjorn Poonen Conjecture (Wang 1961) Wang tiles If a finite set of tiles can tile the plane, there exists a Group theory periodic tiling. F.p. groups Words Word problem Assuming this, Wang gave an algorithm for deciding whether Markov properties a finite set of tiles can tile the plane. Topology Homeomorphism problem Knot theory Algebraic geometry But. . . Varieties Isomorphism problem Theorem (Berger 1967) Commutative algebra F.g. algebras 1. Wang’s conjecture is wrong! Some tile sets can tile the F.g. fields plane only aperiodically. References 2. The problem of deciding whether a given tile set can tile the plane is undecidable.

Undecidability Group theory everywhere Bjorn Poonen Wang tiles Group theory Question F.p. groups Words Word problem Can a computer decide whether an element of a group Markov properties equals the identity? Topology Homeomorphism problem Knot theory To make sense of this question, we must specify Algebraic geometry Varieties Isomorphism problem 1. how the group is described, and Commutative algebra 2. how the element is described. F.g. algebras F.g. fields References The descriptions should be suitable for input into a Turing machine.

Undecidability Finitely presented groups (examples) everywhere Bjorn Poonen Example (Pairs of integers) Wang tiles Group theory Z 2 = � a , b | ab = ba � F.p. groups Words Word problem Think of a as (1 , 0) and b as (0 , 1). Markov properties Topology Homeomorphism Example (The symmetric group on 3 letters) problem Knot theory Algebraic geometry S 3 = � r , t | r 3 = 1 , t 2 = 1 , trt − 1 = r − 1 � . Varieties Isomorphism problem Commutative Think of r as (123) and t as (12). algebra F.g. algebras F.g. fields Example (The free group on 2 generators) References F 2 = � g 1 , g 2 | � . An f.p. group can be described using finitely many characters, and hence is suitable input for a Turing machine.

Undecidability Finitely presented groups (definition) everywhere Bjorn Poonen Wang tiles Group theory Definition F.p. groups Words Word problem A group G is finitely presented (f.p.) if there exist n ∈ N and Markov properties finitely many elements r 1 , . . . , r m ∈ F n such that G ≃ F n / R Topology Homeomorphism where R is the smallest normal subgroup of F n containing problem Knot theory r 1 , . . . , r m . Algebraic geometry Varieties Think of r 1 , . . . , r n as relations imposed on the generators of Isomorphism problem Commutative G , and think of R as the set of relations implied by algebra r 1 , . . . , r n . We write F.g. algebras F.g. fields References G = � g 1 , . . . , g n | r 1 , . . . , r m � .

Undecidability Words everywhere Bjorn Poonen How are elements of f.p. groups represented? Wang tiles Definition Group theory F.p. groups A word in the elements of a set S is a finite sequence in Words which each term is an element s ∈ S or a symbol s − 1 for Word problem Markov properties some s ∈ S . Topology Homeomorphism problem Knot theory Example Algebraic geometry Varieties aba − 1 a − 1 bb − 1 b is a word in a and b . Isomorphism problem Commutative If G is an f.p. group with generators g 1 , . . . , g n , then each algebra F.g. algebras word in g 1 , . . . , g n represents an element of G . F.g. fields References Example In S 3 = � r , t | r 3 = 1 , t 2 = 1 , trt − 1 = r − 1 � with r = (123) and t = (12), the words tr and r − 1 t both represent (23). And trt − 1 r represents the identity.

Undecidability The word problem everywhere Bjorn Poonen Given a f.p. group G , we have Wang tiles Word problem for G Group theory F.p. groups Find an algorithm with Words Word problem input: a word w in the generators of G Markov properties output: YES or NO, according to whether w Topology Homeomorphism problem represents the identity in G. Knot theory Algebraic geometry Varieties Harder problem: Isomorphism problem Commutative algebra Uniform word problem F.g. algebras F.g. fields Find an algorithm with References input: a f.p. group G, and a word w in the generators of G output: YES or NO, according to whether w represents the identity in G.

Undecidability Word problem for F n everywhere Bjorn Poonen Wang tiles Group theory The word problem for the free group F n is decidable: given a F.p. groups word in the generators, it represents the identity if and only Words Word problem if the reduced word obtained by iteratively cancelling Markov properties Topology adjacent inverses is the empty word. Homeomorphism problem Knot theory Example Algebraic geometry Varieties In the free group F 2 = � a , b � , the reduced word associated to Isomorphism problem Commutative aba − 1 bb − 1 abb algebra F.g. algebras F.g. fields References is abbb .

Undecidability Undecidability of the word problem everywhere Bjorn Poonen For any f.p. group G , the set W of words w Wang tiles representing the identity in G is listable: a computer Group theory F.p. groups can generate all possible consequences of the given Words Word problem relations. Markov properties But the word problem for G is asking whether W is Topology Homeomorphism computable, whether an algorithm can test whether a problem Knot theory particular word belongs to W . Algebraic geometry Varieties Isomorphism problem In fact: Commutative algebra Theorem (P. S. Novikov 1955) F.g. algebras F.g. fields There exists an f.p. group G such that the word problem for References G is undecidable. Corollary The uniform word problem is undecidable.

Undecidability Markov properties everywhere Bjorn Poonen Definition Wang tiles Group theory A property of f.p. groups is called a Markov property if F.p. groups Words 1. there exists an f.p. group G 1 with the property, and Word problem Markov properties 2. there exists an f.p. group G 2 that cannot be embedded Topology Homeomorphism in any f.p. group with the property. problem Knot theory Algebraic geometry Example Varieties Isomorphism problem The property of being finite is a Markov property: Commutative algebra 1. There exists a finite group! F.g. algebras F.g. fields 2. The f.p. group Z cannot be embedded in any finite References group. Other Markov properties: trivial, abelian, nilpotent, solvable, free, torsion-free.

Undecidability everywhere Bjorn Poonen Theorem (Adian & Rabin 1955–1958) Wang tiles For each Markov property P , the problem of deciding Group theory F.p. groups whether an arbitrary f.p. group has P is undecidable. Words Word problem Markov properties Sketch of proof. Topology Homeomorphism problem Given an f.p. group G and a word w in its generators, one Knot theory can build another f.p. group K such that K has P if and Algebraic geometry Varieties only if w represents the identity of G . If P were a decidable Isomorphism problem property, then one could solve the uniform word problem. Commutative algebra F.g. algebras F.g. fields Corollary References There is no algorithm to decide whether an f.p. group is trivial.

Undecidability The homeomorphism problem everywhere Bjorn Poonen Wang tiles Group theory F.p. groups Words Word problem Markov properties Question Topology Given two manifolds, can one decide whether they are Homeomorphism problem Knot theory homeomorphic? Algebraic geometry Varieties To make sense of this question, we must specify how a Isomorphism problem manifold is described. The description should be suitable for Commutative algebra input into a Turing machine. F.g. algebras F.g. fields References