A non-embedding result for Thompsons Group V Thompsons Group V - PowerPoint PPT Presentation

A non- embedding result for A non-embedding result for Thompson’s Group V Thompson’s Group V Nathan Corwin Introduction co CF groups Nathan Corwin Wreath Products Thompson’s University of Nebraska – Lincoln Group V Dynamics of V Groups St Andrews 2013 Proof of Main Result 7 August 2013 s-ncorwin1@math.unl.edu

Overview A non- embedding result for Thompson’s Group V Theorem (C. 2013) Nathan Z ≀ Z 2 does not embed into Thompson’s Group V . Corwin Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview A non- embedding result for Thompson’s Group V Theorem (C. 2013) Nathan Z ≀ Z 2 does not embed into Thompson’s Group V . Corwin Introduction co CF groups Give some motivation. Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Overview A non- embedding result for Thompson’s Group V Theorem (C. 2013) Nathan Z ≀ Z 2 does not embed into Thompson’s Group V . Corwin Introduction co CF groups Give some motivation. Wreath Products Define wreath product. Thompson’s Group V Dynamics of V Proof of Main Result

Overview A non- embedding result for Thompson’s Group V Theorem (C. 2013) Nathan Z ≀ Z 2 does not embed into Thompson’s Group V . Corwin Introduction co CF groups Give some motivation. Wreath Products Define wreath product. Thompson’s Define Thompson’s Group V. Group V Dynamics of V Proof of Main Result

Overview A non- embedding result for Thompson’s Group V Theorem (C. 2013) Nathan Z ≀ Z 2 does not embed into Thompson’s Group V . Corwin Introduction co CF groups Give some motivation. Wreath Products Define wreath product. Thompson’s Define Thompson’s Group V. Group V Dynamics of Briefly discuss dynamics in the group. V Proof of Main Result

Overview A non- embedding result for Thompson’s Group V Theorem (C. 2013) Nathan Z ≀ Z 2 does not embed into Thompson’s Group V . Corwin Introduction co CF groups Give some motivation. Wreath Products Define wreath product. Thompson’s Define Thompson’s Group V. Group V Dynamics of Briefly discuss dynamics in the group. V Proof of Main Briefly discuss the proof of the theorem. Result

C F and co C F A non- embedding result for Thompson’s Group V Nathan Corwin Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C F and co C F A non- In the mid-1980’s Muller and Schupp showed that the embedding result for class of all of groups that have a context free word Thompson’s Group V problem (denoted C F ) is equivalent to the the class of Nathan groups that are virtually free. Corwin Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C F and co C F A non- In the mid-1980’s Muller and Schupp showed that the embedding result for class of all of groups that have a context free word Thompson’s Group V problem (denoted C F ) is equivalent to the the class of Nathan groups that are virtually free. Corwin A natural generalization of C F is the class co C F : all Introduction groups for which the coword problem is context free. co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

C F and co C F A non- In the mid-1980’s Muller and Schupp showed that the embedding result for class of all of groups that have a context free word Thompson’s Group V problem (denoted C F ) is equivalent to the the class of Nathan groups that are virtually free. Corwin A natural generalization of C F is the class co C F : all Introduction groups for which the coword problem is context free. co CF groups Wreath This class was first introduced by Holt, Rees, R¨ over, and Products Thomas in 2006. Thompson’s Group V Dynamics of V Proof of Main Result

C F and co C F A non- In the mid-1980’s Muller and Schupp showed that the embedding result for class of all of groups that have a context free word Thompson’s Group V problem (denoted C F ) is equivalent to the the class of Nathan groups that are virtually free. Corwin A natural generalization of C F is the class co C F : all Introduction groups for which the coword problem is context free. co CF groups Wreath This class was first introduced by Holt, Rees, R¨ over, and Products Thomas in 2006. Thompson’s Group V They showed that co C F has many closure properties. Dynamics of Closed under: V direct products; Proof of Main Result standard restricted wreath products where the top group is C F ; passing to finitely generated subgroups; passing to finite index over-groups.

co C F A non- embedding Two conjectures from that paper: result for Thompson’s Group V Nathan Corwin Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

co C F A non- embedding Two conjectures from that paper: result for Thompson’s Group V Nathan Corwin 1 If C ≀ T is in co C F , then T must be in C F ; Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

co C F A non- embedding Two conjectures from that paper: result for Thompson’s Group V Nathan Corwin 1 If C ≀ T is in co C F , then T must be in C F ; Introduction co CF groups My theorem supports this conjecture. Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

co C F A non- embedding Two conjectures from that paper: result for Thompson’s Group V Nathan Corwin 1 If C ≀ T is in co C F , then T must be in C F ; Introduction co CF groups My theorem supports this conjecture. Wreath Products Thompson’s 2 co C F is not closed under free products. Group V Dynamics of V Proof of Main Result

co C F A non- embedding Two conjectures from that paper: result for Thompson’s Group V Nathan Corwin 1 If C ≀ T is in co C F , then T must be in C F ; Introduction co CF groups My theorem supports this conjecture. Wreath Products Thompson’s 2 co C F is not closed under free products. Group V Dynamics of V The leading candidate to show the second conjecture is the Proof of Main Result group Z ∗ Z 2 .

co C F and V A non- embedding result for Thompson’s Group V In 2007 Lehnert and Schweitzer showed that R. Nathan Thompson’s group V is in co C F . Corwin Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

co C F and V A non- embedding result for Thompson’s Group V In 2007 Lehnert and Schweitzer showed that R. Nathan Thompson’s group V is in co C F . Corwin This was a surprising result. Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

co C F and V A non- embedding result for Thompson’s Group V In 2007 Lehnert and Schweitzer showed that R. Nathan Thompson’s group V is in co C F . Corwin This was a surprising result. Introduction It also put into doubt the belief that Z ∗ Z 2 is not in co CF groups co C F as V contains many copies of Z and Z 2 and free Wreath Products products of subgroups are common in V . Thompson’s Group V Dynamics of V Proof of Main Result

co C F and V A non- embedding result for Thompson’s Group V In 2007 Lehnert and Schweitzer showed that R. Nathan Thompson’s group V is in co C F . Corwin This was a surprising result. Introduction It also put into doubt the belief that Z ∗ Z 2 is not in co CF groups co C F as V contains many copies of Z and Z 2 and free Wreath Products products of subgroups are common in V . Thompson’s ıaz showed that Z ∗ Z 2 does Group V In 2009, Bleak and Salazar-D` Dynamics of not embed into V . V Proof of Main Result

co C F and V A non- embedding result for Thompson’s Group V In 2007 Lehnert and Schweitzer showed that R. Nathan Thompson’s group V is in co C F . Corwin This was a surprising result. Introduction It also put into doubt the belief that Z ∗ Z 2 is not in co CF groups co C F as V contains many copies of Z and Z 2 and free Wreath Products products of subgroups are common in V . Thompson’s ıaz showed that Z ∗ Z 2 does Group V In 2009, Bleak and Salazar-D` Dynamics of not embed into V . V In that paper, they conjectured that Z ≀ Z 2 does not Proof of Main Result embed into V .

Other motivation Structure of the R. Thompson’s groups A non- embedding result for Thompson’s Group V Nathan Corwin Richard Thompson discovered F < T < V in 1965. Introduction co CF groups Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result

Other motivation Structure of the R. Thompson’s groups A non- embedding result for Thompson’s Group V Nathan Corwin Richard Thompson discovered F < T < V in 1965. Introduction In 1999, Guba and Sapir showed that Z ≀ Z embeds into F , co CF groups and thus embeds into T and V as well. Wreath Products Thompson’s Group V Dynamics of V Proof of Main Result