Undecidability in group theory, topology, and F.p. groups Word - PowerPoint PPT Presentation

Undecidability in group theory, topology, and analysis Bjorn Poonen Group theory Undecidability in group theory, topology, and F.p. groups Word problem Markov properties analysis Topology Fundamental group Homeomorphism problem Manifold? Bjorn Poonen Knot theory Analysis Inequalities Complex analysis Integration Rademacher Lecture 2 November 7, 2017

Undecidability in Group theory group theory, topology, and analysis Bjorn Poonen Question Group theory F.p. groups Can a computer decide whether Word problem Markov properties two given elements of a group are equal? Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration

Undecidability in Group theory group theory, topology, and analysis Bjorn Poonen Question Group theory Can a computer decide whether F.p. groups Word problem Markov properties two given elements of a group are equal Topology a given element of a group equals the identity? Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration

Undecidability in Group theory group theory, topology, and analysis Bjorn Poonen Question Group theory Can a computer decide whether F.p. groups Word problem Markov properties two given elements of a group are equal Topology a given element of a group equals the identity? Fundamental group Homeomorphism problem Manifold? Knot theory Analysis To make sense of this question, we must specify Inequalities Complex analysis 1. how the group is described Integration 2. how the element is described The descriptions should be suitable for input into a Turing machine.

Undecidability in Group theory group theory, topology, and analysis Bjorn Poonen Question Group theory Can a computer decide whether F.p. groups Word problem Markov properties two given elements of a group are equal Topology a given element of a group equals the identity? Fundamental group Homeomorphism problem Manifold? Knot theory Analysis To make sense of this question, we must specify Inequalities Complex analysis 1. how the group is described: f.p. group Integration 2. how the element is described: word The descriptions should be suitable for input into a Turing machine.

Undecidability in Example: The symmetric group S 3 group theory, topology, and analysis Bjorn Poonen Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis In cycle notation, r = (123) and t = (12). These satisfy Integration r 3 = 1 , t 2 = 1 , trt − 1 = r − 1 It turns out that r and t generate S 3 , and every relation involving them is a consequence of the relations above: S 3 = � r , t | r 3 = 1 , t 2 = 1 , trt − 1 = r − 1 � .

Undecidability in Finitely presented groups group theory, topology, and analysis Bjorn Poonen Definition Group theory F.p. groups An f.p. group is a group specified by finitely many Word problem Markov properties generators and finitely many relations. Topology Fundamental group Homeomorphism problem Example Manifold? Knot theory Analysis Z × Z = � a , b | ab = ba � Inequalities Complex analysis Integration Example The free group on 2 (noncommuting) generators is F 2 := � a , b | �

Undecidability in Representing elements of an f.p. group: words group theory, topology, and analysis Bjorn Poonen Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism S 3 = � r , t | r 3 = 1 , t 2 = 1 , trt − 1 = r − 1 � . problem Manifold? Knot theory Analysis Definition Inequalities Complex analysis A word is a sequence of the generator symbols and their Integration inverses, such as tr − 1 ttrt − 1 rrr . Since r and t generate S 3 , every element of S 3 is represented by a word, but not necessarily in a unique way. Example The words tr and r − 1 t both represent (23).

Undecidability in The word problem group theory, topology, and analysis Given an f.p. group G , we have Bjorn Poonen Word problem for G Group theory F.p. groups Find an algorithm with Word problem Markov properties input: a word w in the generators of G Topology Fundamental group output: YES or NO, according to whether w = 1 in G. Homeomorphism problem Manifold? Knot theory Analysis Harder problem: Inequalities Complex analysis Integration Uniform word problem Find an algorithm with input: an f.p. group G, and a word w in the generators of G output: YES or NO, according to whether w = 1 in G.

Undecidability in Word problem for F n group theory, topology, and analysis Theorem Bjorn Poonen The word problem for the free group F n is decidable. Group theory Algorithm to decide whether a given word w represents 1: F.p. groups Word problem Markov properties 1. Repeatedly cancel adjacent inverses until there is Topology nothing left to cancel. Fundamental group Homeomorphism 2. Check if the end result is the empty word. problem Manifold? Knot theory Example Analysis Inequalities Complex analysis In the free group F 2 = � a , b � , given the word Integration aba − 1 bb − 1 abb , cancellation leads to abbb , which is not the empty word, so aba − 1 bb − 1 abb does not represent the identity.

Undecidability in Undecidability of the word problem group theory, topology, and analysis Bjorn Poonen Theorem (P. S. Novikov and Boone, Group theory F.p. groups independently in the 1950s) Word problem Markov properties There exists an f.p. group G such that the word problem for Topology Fundamental group G is undecidable. Homeomorphism problem Manifold? The strategy of the proof, as for Hilbert’s tenth problem, is Knot theory to build a group G such that solving the word problem for G Analysis Inequalities is at least as hard as solving the halting problem. Complex analysis Integration Corollary The uniform word problem is undecidable.

Undecidability in Markov properties group theory, topology, and analysis Bjorn Poonen Definition Group theory F.p. groups A property of f.p. groups is called a Markov property if Word problem Markov properties 1. there exists an f.p. group G 1 with the property, and Topology Fundamental group 2. there exists an f.p. group G 2 that cannot be embedded Homeomorphism problem in any f.p. group with the property. Manifold? Knot theory Analysis Example Inequalities Complex analysis Integration The property of being finite is a Markov property, because 1. There exists a finite group! 2. Z cannot be embedded in any finite group. Other Markov properties: trivial, abelian, free, . . . .

Undecidability in group theory, topology, and Theorem (Adian & Rabin 1955–1958) analysis Bjorn Poonen For each Markov property P , the problem of deciding whether an arbitrary f.p. group has P is undecidable. Group theory F.p. groups Word problem Markov properties Sketch of proof. Topology Embed the uniform word problem in this P problem: Fundamental group Homeomorphism problem Given an f.p. group G and a word w in its generators, Manifold? Knot theory build another f.p. group K such that Analysis Inequalities K has P ⇐ ⇒ w = 1 in G . Complex analysis Integration Example There is no algorithm to decide whether an f.p. group is trivial.

Undecidability in Fundamental group group theory, topology, and Fix a manifold M . analysis Bjorn Poonen Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration

Undecidability in Fundamental group group theory, topology, and Fix a manifold M and a point p . analysis Bjorn Poonen Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration

Undecidability in Fundamental group group theory, topology, and Fix a manifold M and a point p . analysis Consider paths in M that start and end at p . Bjorn Poonen Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration

Undecidability in Fundamental group group theory, topology, and Fix a manifold M and a point p . analysis Consider paths in M that start and end at p . Bjorn Poonen Paths are homotopic if one can be deformed to the other. Group theory F.p. groups Word problem Markov properties Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration

Undecidability in Fundamental group group theory, topology, and Fix a manifold M and a point p . analysis Consider paths in M that start and end at p . Bjorn Poonen Paths are homotopic if one can be deformed to the other. Group theory F.p. groups Fundamental group π 1 ( M ) := { paths } / homotopy. Word problem Markov properties Group law: concatenation of paths. Topology Fundamental group Homeomorphism problem Manifold? Knot theory Analysis Inequalities Complex analysis Integration