Delzant's T-invariant, Kolmogorov complexity and one-relator groups

We prove that ``almost generically'' for a one-relator group Delzant's $T$-invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator groups and on the methods of the theory of Kolmogorov-Chaitin complexity. We also give a precise asymptotic estimate (when $k$ is fixed and $n$ goes to infinity) for the number $I_{k,n}$ of isomorphism classes of $k$-generator one-relator groups with a cyclically reduced defining relator of length $n$: $$I_{k,n}\sim \frac{(2k-1)^n}{nk!2^{k+1}}.$$ Here $f(n)\sim g(n)$ means that $\lim_{n\to\infty} f(n)/g(n)=1$.Comment: A revised version, to appear in Comment. Math. Hel

The complexity of topological group isomorphism

We study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound

An Efficient Quantum Algorithm for some Instances of the Group Isomorphism Problem

In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group $A$ by a cyclic group $Z_m$ with the order of $A$ coprime with $m$. More precisely, the running time of that algorithm is almost linear in the order of the input groups. In this paper we present a quantum algorithm solving the same problem in time polynomial in the logarithm of the order of the input groups. This algorithm works in the black-box setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms.Comment: 20 pages; this is the full version of a paper that will appear in the Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010

Automorphisms of the mapping class group of a nonorientable surface

Let $S$ be a nonorientable surface of genus $g\ge 5$ with $n\ge 0$ punctures, and \Mcg(S) its mapping class group. We define the complexity of $S$ to be the maximum rank of a free abelian subgroup of \Mcg(S). Suppose that $S_1$ and $S_2$ are two such surfaces of the same complexity. We prove that every isomorphism \Mcg(S_1)\to\Mcg(S_2) is induced by a diffeomorphism $S_1\to S_2$. This is an analogue of Ivanov's theorem on automorphisms of the mapping class groups of an orientable surface, and also an extension and improvement of the first author's previous result.Comment: 21 pages, 10 figures, revision and corrections, to appear in Geometriae Dedicat

An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the $k$-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page

Combinatorial refinement on circulant graphs

The combinatorial refinement techniques have proven to be an efficient approach to isomorphism testing for particular classes of graphs. If the number of refinement rounds is small, this puts the corresponding isomorphism problem in a low-complexity class. We investigate the round complexity of the 2-dimensional Weisfeiler-Leman algorithm on circulant graphs, i.e. on Cayley graphs of the cyclic group $\mathbb{Z}_n$, and prove that the number of rounds until stabilization is bounded by $\mathcal{O}(d(n)\log n)$, where $d(n)$ is the number of divisors of $n$. As a particular consequence, isomorphism can be tested in NC for connected circulant graphs of order $p^\ell$ with $p$ an odd prime, $\ell>3$ and vertex degree $\Delta$ smaller than $p$. We also show that the color refinement method (also known as the 1-dimensional Weisfeiler-Leman algorithm) computes a canonical labeling for every non-trivial circulant graph with a prime number of vertices after individualization of two appropriately chosen vertices. Thus, the canonical labeling problem for this class of graphs has at most the same complexity as color refinement, which results in a time bound of $\mathcal{O}(\Delta n\log n)$. Moreover, this provides a first example where a sophisticated approach to isomorphism testing put forward by Tinhofer has a real practical meaning.Comment: 19 pages, 1 figur