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Course, academic year 2023/2024
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Combinatorial Group Theory - NALG033
Title: Kombinatorická teorie grup
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: winter
E-Credits: 9
Hours per week, examination: winter s.:2/2, C [HT]
summer s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Mgr. Pavel Růžička, Ph.D.
Classification: Mathematics > Algebra
Pre-requisite : NALG017
Interchangeability : NMAG432
Is incompatible with: NMAG432, NMAG431
Is interchangeable with: NMAG432, NMAG431
Annotation -
Last update: G_M (05.10.2001)
Winter term: Subgroups of free groups (Nielsen's and Reidemaister's method), Tietze transformations, HNN extensions, free products with an amalgamated subgroup, geometrical methods, Cayley complexes. Spring term: Other selected topics in elementary combinatorical group theory.
Literature - Czech
Last update: RNDr. Pavel Zakouřil, Ph.D. (05.08.2002)

1. R.C. Lyndon, P.E. Schupp, Combinatorial Group Theory, Springer Verlag 1977

2. W. Magnus, A. Karras, D. Solitar, Combinatorial Group Theory: Presentations of groups in terms of generators and relations, Interscience Publishers, John Willey Sons, 1966; ruský překlad: Kombinatornaja teorija grupp: Predstavlenije grupp v terminach obrazujuščich i sootnošenij, Nauka, Moskva 1974

Syllabus -
Last update: T_KA (17.05.2004)

Winter semester :

1. Free group, subgroups of a free group (the method of Nielsen and Reidemeister), the relationship between the index and the rank of subgroup of a group

of a finite index, subgroups of finite rank as free factors in a subgroup of a finite index. Conjugation and cyclically reduced words.

2. Tietze transformations.

3. HNN extensions, defining relations, Britton's lemma and the normal form theorem, applications of HNN extensions.

4. Free products with an amalgamated subgroup, defining relations, the normal form theorem.

5. Geometrical methods, the fundamental group of a two-dimensional complex, application for a geometrical proof that a subgroup of a free group is free,

Kurosh's theorem, Grushko -- von Neumann's theorem.

6. Cayley complexes

Summer semester :

According to an interest, some of the following topics will be tought.

1. Higman's embedding theorem.

2. Small cancellation theory.

3. Braid group, the word problem, factors, connections to authomorphisms of free groups.

4. Groups acting on trees.

5. Hyperbolic groups.

6. Tessellations and Fuchsian complexes.

7. Solvability of the word problem for groups with one defining relation.

8. Bipolar structures.

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