Algorithmische Gruppentheorie

Organisatorisches

Dozent: Volker Diekert

Übungen: Lukas Fleischer

Zeit	Raum	Termine
Di 14:00-15:30	0.124	wöchentlich
Do 11:30-13:00	0.124	wöchentlich

Inhalt

Bereits 1911 formulierte Max Dehn drei fundamentale algorithmische Probleme in der (kombinatorischen) Gruppentheorie:

Wortproblem: Ist ein gegebenes Gruppenelement (als Wort in Erzeugern) das Einselement in der Gruppe?
Konjugationsproblem: Sind zwei Elemente konjugiert?
Isomorphieproblem: Definieren zwei gegebene Darstellungen isomorphe Gruppen?

Im Allgemeinen sind alle diese Fragen unentscheidbar, also kann man positive Antworten nur in Spezialfällen erhalten. Die weitreichensten Ergebnisse liegen für das Wortproblem vor. Hier gibt es eine große Klasse von Gruppen, die in der Praxis auftreten und für die man sehr gute Algorithmen kennt. In der Vorlesung sollen Techniken behandelt werden, die zu positiven Lösungen zu den obigen Fragen führen und für welche Klasse von Gruppen diese anwendbar sind. Eine prominente Rolle spielen hierbei konfluente Wortersetzungssysteme, die auch in anderen Bereichen zum Einsatz kommen. Insgesamt lebt die Theorie von Querbezügen zu vielen anderen Bereichen, wie Kombinatorik, Topologie, Geometrie, theoretischer Informatik. Dieses Zusammenspiel verschiedener Methoden macht die algorithmische Gruppentheorie sehr attraktiv.

Folien und Skript zur Vorlesung

Folien vom Wintersemester 2016/17 (Stand: 08.02.2017)
Folien zu Power Circuits (Stand: 02.02.2017)
Zusatznotizen (Stand: 22.12.2016)
Folien vom Wintersemester 2014/15
Folien vom Wintersemester 2008/09

Übungsblätter

Blatt 1 (Besprechung am 10.11.2016)
Blatt 2 (Besprechung am 15.11.2016)
Blatt 3 (Besprechung am 06.12.2016 und am 08.12.2016)
Blatt 4 (Besprechung am 10.01.2017, Stand: 10.01.2016)

Literatur

Volker Diekert, Manfred Kufleitner, Gerhard Rosenberger: Diskrete algebraische Methoden, Walter de Gruyter, 2013.
Lyndon, Schupp: Combinatorial Group Theory, Springer, 1977.

News

[Jun’23] The paper “Parallel algorithms for power circuits and the word problem of the Baumslag group” by Caroline Mattes and Armin Weiß has been accepted at Computational Complexity.

[Oct’22] The paper “Lower Bounds for Sorting 16, 17, and 18 Elements” by Florian Stober and Armin Weiß has been accepted at ALENEX 2023.

[Sep’22] The paper “Conelikes and Ranker Comparisons” by Viktor Henriksson and Manfred Kufleitner has been accepted at LATIN 2022.

[Sep’22] The paper “Improved Parallel Algorithms for Generalized Baumslag Groups” by Caroline Mattes and Armin Weiß has been accepted at LATIN 2022.

[Apr’22] The paper “Reachability Games and Parity Games” by Volker Diekert and Manfred Kufleitner has been accepted at ICTAC 2022.

[Apr’22] The paper “Satisfiability Problems for Finite Groups” by Pawel M. Idziak, Piotr Kawalek, Jacek Krzaczkowski and Armin Weiß has been accepted at ICALP 2022.

All news

[Mar’22] The paper “The Power Word Problem in Graph Products” by Florian Stober and Armin Weiß was accepted at DLT 2022.

[Nov’20] Volker Diekert is Partner Investigator in the Australian ARC grant “Geodetic groups: foundational problems in algebra and computer science” at University of Technology Sydney.

[Apr’20] The paper “Groups with ALOGTIME-hard word problems and PSPACE-complete circuit value problems” by Laurent Bartholdi, Michael Figelius, Markus Lohrey and Armin Weiß has been accepted at CCC 2020.

[Apr’20] The paper “Hardness of equations over finite solvable groups under the exponential time hypothesis” by Armin Weiß has been accepted at ICALP 2020.

[Dec’19] The paper “An Automaton Group with PSPACE-Complete Word Problem” by Jan Philipp Wächter and Armin Weiß has been accepted at STACS 2020.

[Nov’19] Carlos Camino was awarded the stuvus Special Prize for exceptional commitment in teaching.

[Jun’19] The paper “The power word problem” by Markus Lohrey and Armin Weiß has been accepted at MFCS 2019.

[May’19] The paper “On the Average Case of MergeInsertion” by Florian Stober and Armin Weiß has been accepted at IWOCA 2019.

[Oct’18] The paper “Worst-Case Efficient Sorting with QuickMergesort” by Stefan Edelkamp and Armin Weiß has been accepted at ALENEX 2019.

[Jun’18] At CCC 2018, Lukas Fleischer received a Best Student Paper Award for his submission “On the Complexity of the Cayley Semigroup Membership Problem”.

[Jun’18] The paper “Testing Simon’s congruence” by Lukas Fleischer and Manfred Kufleitner was accepted at MFCS 2018.

[Jun’18] The paper “The Intersection Problem for Finite Semigroups” by Lukas Fleischer was accepted at DLT 2018.

[Apr’18] The paper “The isomorphism problem for finite extensions of free groups is in PSPACE” by Géraud Sénizergues and Armin Weiß was accepted at ICALP 2018.

[Apr’18] The paper “On the Complexity of the Cayley Semigroup Membership Problem” by Lukas Fleischer was accepted at CCC 2018.

[Jan’18] On March 24-29, 2019 Volker Diekert, Markus Lohrey, Olga Kharlampovich and Alexei Miasnikov will organize the Schloss Dagstuhl Seminar “Algorithmic Problems in Group Theory”.

[Dec’17] The paper “The Intersection Problem for Finite Monoids” by Lukas Fleischer and Manfred Kufleitner was accepted at STACS 2018.

[Jun’17] At the 12th International Computer Science Symposium in Russia (CSR), Lukas Fleischer and Manfred Kufleitner received a Best Paper Award for their publication “Green’s Relations in Finite Transformation Semigroups”, and Armin Weiss received a Best Paper Award for “The conjugacy problem in free solvable groups and wreath product of abelian groups is in $\text{TC}^0$ “ which is joint work with Alexei Miasnikov and Svetla Vassileva.