Schedule:

We are meeting in BA6180 from September 10 to December 3.

**About:**

- I will try to cover the backbone from chapters 2 to 6 of the main book. If possible I will talk about chapter 7, but I am not sure we will be able to talk too much about.

**Bibliography:**

[1] *Buildings.* Peter Abramenko, Kenneth S. Brown. Springer, Graduate Texts in Mathematics, Vol. 248

**Logbook:**

Date | What did we do? | Notes |

September 10 | We talked about properties of pairs (W, S). We defined the deletion, exchange and foldingconditions and discussed parabolic subgroups.[1] Sections 2.3.1 and 2.3.2.See also [1] section 1.5 | N/A |

September 17 | We introduced the action and the coxeter conditions and defined Coxeter Group. We defined what the associated Coxeter Complex is and saw an example of it.[1] Section 2.1, 2.3.3, 2.4, 3.1See section 2.2 to see examples. | |

September 24 | We finished the topics in the notes of previous time: we proved the properties of the Coxeter Complex, [1] Theorem 3.5. We also defined the components associated to an automorphism and discussed why they satisfy the conditions they do: [1] Lemma 3.31.[1] Section 3.1, 3.3.1Our discussion about the reason why the complex is thin was confusing. Here is a written explicit proof: THIN. | |

October 1 | We started the notes of September 24 but got stuck with the proof of Lemma 1 for the whole lecture since we had a whole confusion of what chamber maps should or should not do. To explain the idea of the part of the proof we got confused with I did the attached sketch. [1] Section 3.3, Lemma 3.31. | |

October 8 | This class got cancelled | N/A |

October 15 | We discussed foldings and their properties via examples, finishing the notes of september 24. We did not go over the proofs of the results, just talked about them. [1] Section 3.3.2., 3.3.3. We defined buildings, saw examples and stated B2, B2′, B2” are equivalent. [1] Section 4.1 up to proposition 4.6Originally my idea was to discuss [1] Section 3.4 and sketch the proof of Theorem 3.65, but due to time I decided to skip this. | |

October 22 | We proved B2, B2′ and B2” are equivalentand then discussed what a link is in a simplicial complex. We discussed that links are coxeter simplices again, but got confused about some issues of correspondences. We did not prove the corresponding fact for buildings (which is proposition 4.9).[1] Section 4.1 up to proposition 4.6 | N/A |

October 29 | We defined what B, N, T mean given an action of a group in a building and studied from there: strong transitive actions, Weyl Transitive Actions and introduced in the context of simplicial complexes the Weyl Distance function and with it we proved the Bruhat Decomposition. We did not see the part of subbuildings. [1] Section 6.1.1, but I avoided using the notion of subbuilding in the class, but I introduced it in the notes.[1] Section 6.1.3 up to Corollary 6.12.[1] Section 6.1.4 up to Theorem 6.17 (this is the Bruhat Decomposition).[1] Section 3.5, page 145 (al after definition 3.86) and then section 4.8 up to proposition 4.84. | |

November 5 | Because it’s reading week we made a review of all the main definitions and results we have studied so far. In the remaining time we defined W-metric buildings and Weyl distances, and discussed some examples. [1] Section 5.1, up to Definition 5.2. | |

November 12 | We did the first three pages of the notes up to the triangle inequality, but we didn’t prove this. During the lecture we discussed how (WD2) different options sw or w can occur: basically the weyl distance doesnt change if you take the “corresponding” next chamber but in another appartment. An example where this happens is the buildings obtained from K_{3, 3} (bipartite graph) with every four cycle being an appartment. To see the phenomenon occuring pick a chamber and two opposing ones, but in different appartments. The notes do not have the proof of Lemma 5.5 but we discussed it in class. [1] Section 5.1, and Section 5.3.1 | |

November 19 | We almost finished the notes of previous time. We stopped after defining what an isometry is. [1] Sections 5.4.1, 5.4.2, 5.5.1 | N/A |

November 26 | We finished the notes of previous time by studying different equivalences of isometries and then studied how this allowed us to detect subbuildging as images of buildings under isometric maps. We studied the idea of projections and saw some examples using K_{3, 3} and discussed a bit of the structure of the proof that will allow us to find apartments inside a given building. [1] Sections 5.5.1, 5.3.2 | N/A |

December 3 | I forgot my notes! We were going to see [1] Section 5.5.3, which proves the existence of apartments.Instead we studied the construction of the building of for a prime p. | N/A |

**Notes:**

- The book [1], for U of T students, can be obtained as a MyCopy book in springer through the UofT libraries.
**Chapter 1**of [1] covers finite reflection groups, which are the objects we are trying to generalize to build coxeter systems. From them we will build buildings. It is very instructive to review this chapter as we progress and see how it is done in the case of finite W.- Some sections we have not studied at all but that if we had more time we would study are the following ones:
- Section 2.5. In here the notion of root, hyperplanes and “killing” form appear to construct groups from a given Coxeter matrix via the canonical representation.
- Section 4.3 to 4.7. Its basically the core of the theory of buildings from simplicial perspective, and it is a monstruosity we are not seeing it, but because we want to highlight what happens in chapter 6 we just return to them when needed.

**Alternative References:**

# | Reference: |

1 | An application of the bulding to orbital integralsJonathan D. Rogawski |

2 | Twin Buildings and Application to S-Arithmetic GroupsPeter Abramenko |

3 | Buildings and their applications in Geometry and TopologyLizhen Ji |

4 | Lectures on Buildings, Updated and Revised editionMark Ronan |