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On the complexity of intersection and conjugacy problems in free groups

J. Avenhaus, K. Madlener
1984 Theoretical Computer Science  
Having a Nielsen reduced set of generators for subgroups H and K one can solve a Ict of intersectlon and conjugacy problems in polynomial time in a uniform way.  ...  Nielsen type arguments have been used to prove some problems in free group? (e.g., the generalized word problem) [2] to be P-complete. In this paper we extend this approach.  ...  As by-product a new proof is given that the free groups have the Howson property. The second problem is the conjugacy problem of Moldavanskii.  ... 
doi:10.1016/0304-3975(84)90046-x fatcat:qkdvbm6v5faupp6luyogzhuy5y

Detecting conjugacy stability of subgroups in certain classes of groups [article]

Isabel Fernández Martínez, Denis Serbin
2020 arXiv   pre-print
In this paper we consider the conjugacy stability property of subgroups and provide effective procedures to solve the problem in several classes of groups.  ...  In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable.  ...  We would like to thank Alexei Miasnikov for suggesting to take a look at the conjugacy stability problem in free groups.  ... 
arXiv:2007.07490v1 fatcat:3ourw2ayy5byrmhy74rdkyevnm

Computational complexity and the conjugacy problem

Alexei Miasnikov, Paul Schupp
2017 Computability - The Journal of the Assosiation  
We also consider the complexity relationship of the "half-conjugacy" problem to the conjugacy problem.  ...  The conjugacy problem for a finitely generated group G is the two-variable problem of deciding for an arbitrary pair (u,v) of elements of G, whether or not u is conjugate to v in G.  ...  The complexity of an individual conjugacy problem depends on the fixed element u 0 , and its complexity may be much less than that of the conjugacy problem in G.  ... 
doi:10.3233/com-160060 fatcat:h3c3ebcaunaedom4czsgyw3a2q

Page 6176 of Mathematical Reviews Vol. , Issue 93k [page]

1993 Mathematical Reviews  
a conjugacy of the two groups in Isom(H”).  ...  For a 2-complex K one obtains a presentation Px of its funda- mental group in the standard way: the generators correspond to the 1-cells in the complement of a maximal tree in the 1-skeleton, and the relations  ... 

Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)

Volker Diekert, Olga Kharlampovich, Markus Lohrey, Alexei Myasnikov, Michael Wagner
2019 Dagstuhl Reports  
On one hand, groups played an important rule in many developments in complexity theory and automata theory.  ...  On the other hand, concepts from these computer science fields as well as efficient algorithms, cryptography, and data compression led to the formulation of new questions in group theory.  ...  Zametki, 25(6):811-824, 956, 1979. 3.5 On the intersection problem for free-abelian by free groups Jordi Delgado Rodríguez and Enric Ventura Capell Main reference Jordi Delgado, "Extensions of Free Groups  ... 
doi:10.4230/dagrep.9.3.83 dblp:journals/dagstuhl-reports/DiekertKLM19 fatcat:7b3cwlpo6zdsxfzhiznob5cvsi

Polynomial time conjugacy in wreath products and free solvable groups [article]

Svetla Vassileva
2010 arXiv   pre-print
We prove that the complexity of the Conjugacy Problems for wreath products and for free solvable groups is decidable in polynomial time.  ...  Using this result and properties of the Magnus embedding, we show that the Conjugacy and Conjugacy Search Problems in free solvable groups are computable in polynomial time.  ...  Introduction In this paper we attack the computational complexity of the Conjugacy and Conjugacy Search Problems in free solvable groups.  ... 
arXiv:1011.5931v1 fatcat:jrakkcgcdfawlffzu42n3t4k34

Orbit decidability and the conjugacy problem for some extensions of groups [article]

O. Bogopolski, A. Martino, E. Ventura
2007 arXiv   pre-print
On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy  ...  Also, we give an easy way of constructing groups of the form Z^4 F_n and F_3 F_n with unsolvable conjugacy problem.  ...  We formulate the question for free and free abelian groups because, if one allows an arbitrary ambient F , then the answer is negative: consider Collins-Miller example of a finitely presented group G with  ... 
arXiv:0712.3104v1 fatcat:q6asgc5kv5a6jcgslere23vxg4

Orbit decidability and the conjugacy problem for some extensions of groups

O. Bogopolski, A. Martino, E. Ventura
2009 Transactions of the American Mathematical Society  
On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy  ...  Also, we give an easy way of constructing groups of the form Z 4 Fn and F 3 Fn with unsolvable conjugacy problem.  ...  The first named author is partially supported by the INTAS grant N 03-51-3663 and by the grant Complex integration projects of SB RAS N 1.9.  ... 
doi:10.1090/s0002-9947-09-04817-x fatcat:dhsqfsfhtremzksekbahztfpza

Page 3982 of Mathematical Reviews Vol. , Issue 86i [page]

1986 Mathematical Reviews  
(D-KSRL); Madlener, K. 86i:20053 (D-KSRL) On the complexity of intersection and conjugacy problems in free groups. Theoret. Comput. Sci. 32 (1984), no. 3, 279-295.  ...  Having a Nielsen reduced set of generators for subgroups H and K, one can solve a lot of intersection and conjugacy problems in polynomial time in a uniform way.  ... 

Integration and conjugacy in knot theory [article]

Iain Moffatt
2005 arXiv   pre-print
In particular we study the relationship between known solutions from small cancellation theory and normal surface theory for the word and conjugacy problems of the groups of (prime) alternating links.  ...  This thesis consists of three self-contained chapters. The first two concern quantum invariants of links and three manifolds and the third contains results on the word problem for link groups.  ...  It is the free product of the Dehn presentation of L and the infinite cyclic group. Therefore, solving the conjugacy (and word) problem for the augmented link group solves it for the link group.  ... 
arXiv:math/0511432v1 fatcat:yyae4k57pfec7gqyj7ej2fkftm

Page 1111 of Mathematical Reviews Vol. 55, Issue 4 [page]

1978 Mathematical Reviews  
A group G is the direct sum of the R-complexes R,(i € J) if G is the set-theoretical union of the R,’s and, for each i € J, R, intersected with the join (in Lg) of the remaining ones is Z.  ...  The first group is described in the notation of the second author [On group theoretic problems and their classification, Ann. of Math.  ... 

Page 4471 of Mathematical Reviews Vol. , Issue 83k [page]

1983 Mathematical Reviews  
P. 83k:20036 Estimates of the degree of complexity of an algorithm for solving the word problem and the conjugacy problem for a class of groups.  ...  The author, by constructing a suitable Turing machine, gives upper bounds for the degree of complexity of Dehn’s algorithms for the word and conjugacy problems for small cancellation groups.  ... 

THE CONJUGACY PROBLEM IN AMALGAMATED PRODUCTS I: REGULAR ELEMENTS AND BLACK HOLES

ALEXANDRE V. BOROVIK, ALEXEI G. MYASNIKOV, VLADIMIR N. REMESLENNIKOV
2007 International journal of algebra and computation  
We discuss the time complexity of the word and conjugacy problems for free products G = A C B of groups A and B with amalgamation over a subgroup C.  ...  We stratify the set of elements of G with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable  ...  The third author was supported by EPSRC grant GR/R29451 and by RFFI grants 02-01-00192 and 05-01-00057.  ... 
doi:10.1142/s0218196707003652 fatcat:67q53fzxlrhchfrc5jw3mmbjxm

Morse theory and conjugacy classes of finite subgroups [article]

Noel Brady, Matt Clay, Pallavi Dani
2007 arXiv   pre-print
We also construct examples of groups of type F_n inside mapping class groups, Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order elements.  ...  Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups.  ...  solves Problem 3.10 in [9] ), in Aut(F k ), and Out(F k ), where the free group has sufficiently high rank.  ... 
arXiv:0709.3802v1 fatcat:el6lgq22pragpi5ascg374t54u

Key-agreement based on automaton groups [article]

Rostislav Grigorchuk, Dima Grigoriev
2019 arXiv   pre-print
subgroup of the affine group with the unsolvable conjugacy problem  ...  We suggest several automaton groups as key-agreement platforms for Anshl-Anshel-Goldfeld metascheme, they include Grigorchuk and universal Grigorchuk groups, Hanoi 3-Towers group, Basilica group and a  ...  ) and is grateful to Max-Planck Institut fuer Mathematik, Bonn during staying in which the paper was conceived.  ... 
arXiv:1905.03124v1 fatcat:ionugguliba4zhwlt22vcb5imq
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