|Thesis title in Czech:||Bruhatovy-Titsovy budovy|
|Thesis title in English:||Bruhat-Tits buildings|
|Key words:||Bruhatovy-Titsovy budovy, speciální lineární grupa nad p-adickými čísly, grafová vzdálenost v bytech|
|English key words:||Bruhat-Tits building, special linear group over p-adic numbers, graph distance in apartments|
|Academic year of topic announcement:||2015/2016|
|Type of assignment:||diploma thesis|
|Department:||Department of Algebra (32-KA)|
|Supervisor:||Mgr. Vítězslav Kala, Ph.D.|
|Author:||hidden - assigned and confirmed by the Study Dept.|
|Date of registration:||10.05.2016|
|Date of assignment:||16.05.2016|
|Confirmed by Study dept. on:||16.06.2016|
|Date and time of defence:||13.06.2017 00:00|
|Date of electronic submission:||12.05.2017|
|Date of submission of printed version:||12.05.2017|
|Date of proceeded defence:||13.06.2017|
|Given a reductive group G over a local non-archimedean field F, Bruhat and Tits defined its building B as a certain (infinite) simplicial complex that carries a lot of information about the structure of G (e.g., it can be used to parametrize norms on G or filtrations of G(O_F) by compact subgroups).
The goal of the thesis is to study buildings B of specific groups G, to explicitly describe B and the corresponding filtrations, and to determine geometrical and combinatorial properties of B (e.g., by giving formulas for the number of points in balls). Already the case when G is split classical is very non-trivial, but it would be even more interesting to compare this with the non-split situation. Time permitting, the student may then focus on some of the applications of buildings, for example to representation theory.
Among other things, the thesis will require learning the basics of linear algebraic groups and p-adic fields.
| Tits, Jacques: Reductive groups over local fields. Automorphic forms, representations and L-functions, 29–69, Proc. Sympos. Pure Math., XXXIII, 1979.
 Yu, Jiu-Kang: Bruhat-Tits theory and buildings. Ottawa lectures on admissible representations of reductive p-adic groups, 53–77, Fields Inst. Monogr., 26, 2009.
 Rabinoff, Joseph: The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory. Senior thesis at Harvard University, 2005.
 Brown, Kenneth S.: Buildings. Springer-Verlag, 1989.
 Springer, T. A.: Linear algebraic groups. Second edition. Progress in Mathematics, 9. Birkhäuser Boston, 1998.