TY - GEN
T1 - Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem
AU - Diekert, Volker
AU - Myasnikov, Alexei G.
AU - Weiß, Armin
PY - 2015/6/24
Y1 - 2015/6/24
N2 - In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs: In this work we examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product G = H ∗A K with [H : A] ≥ [K : A] ≥ 2, the Schreier graph with respect to H or K turns out to be non-amenable if and only if [H : A] ≥ 3. Moreover, for an HNN extension of the form G = < H,b | bab-1 = '(a); a 2 A , we show that the Schreier graph of G with respect to the subgroup H is non-amenable if and only if A 6= H 6= '(A). As application of these characterizations we show that under certain conditions the conjugacy problem in fundamental groups of finite graphs of groups with free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield another proof that the set where the conjugacy problem of the Baumslag group G1;2 is decidable in polynomial time is also strongly generic.
AB - In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs: In this work we examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product G = H ∗A K with [H : A] ≥ [K : A] ≥ 2, the Schreier graph with respect to H or K turns out to be non-amenable if and only if [H : A] ≥ 3. Moreover, for an HNN extension of the form G = < H,b | bab-1 = '(a); a 2 A , we show that the Schreier graph of G with respect to the subgroup H is non-amenable if and only if A 6= H 6= '(A). As application of these characterizations we show that under certain conditions the conjugacy problem in fundamental groups of finite graphs of groups with free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield another proof that the set where the conjugacy problem of the Baumslag group G1;2 is decidable in polynomial time is also strongly generic.
KW - Amalgamated product
KW - Amenability
KW - Conjugacy problem
KW - Generic case complexity
KW - HNN extension
KW - Schreier graph
UR - http://www.scopus.com/inward/record.url?scp=84957665972&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84957665972&partnerID=8YFLogxK
U2 - 10.1145/2755996.2756644
DO - 10.1145/2755996.2756644
M3 - Conference contribution
AN - SCOPUS:84957665972
T3 - Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC
SP - 141
EP - 148
BT - ISSAC 2015 - Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation
T2 - 40th ACM International Symposium on Symbolic and Algebraic Computation, ISSAC 2015
Y2 - 6 July 2015 through 9 July 2015
ER -