Magnus subgroups of one-relator surface groups

A one-relator surface group is the quotient of an orientable surface group by the normal closure of a single relator. A Magnus subgroup is the fundamental group of a suitable incompressible sub-surface. A number of results are proved about the intersections of such subgroups and their conjugates, analogous to results of Bagherzadeh, Brodskii, and Collins in classical one-relator group theory.Comment: 15 pages, 3 figure

L^2-Betti numbers of one-relator groups

We determine the L^2-Betti numbers of all one-relator groups and all surface-plus-one-relation groups (surface-plus-one-relation groups were introduced by Hempel who called them one-relator surface groups). In particular we show that for all such groups G, the L^2-Betti numbers b_n^{(2)}(G) are 0 for all n>1. We also obtain some information about the L^2-cohomology of left-orderable groups, and deduce the non-L^2 result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free.Comment: 18 pages, version 3, minor changes. To appear in Math. An

Economical adjunction of square roots to groups

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

Non-orientable surface-plus-one-relation groups

Recently Dicks-Linnell determined the $L^2$-Betti numbers of the orientable surface-plus-one-relation groups, and their arguments involved some results that were obtained topologically by Hempel and Howie. Using algebraic arguments, we now extend all these results of Hempel and Howie to a larger class of two-relator groups, and we then apply the extended results to determine the $L^2$-Betti numbers of the non-orientable surface-plus-one-relation groups.Comment: Version 1. 37 pages. 0 figures. Dedicated to the memory of Karl Gruenber

A generalization of Schur functions: Applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Recent work on return properties of quantum walks (the quantum analogue of random walks) has identified their generating functions of first returns as Schur functions. This is connected with a representation of Schur functions in terms of the operators governing the evolution of quantum walks, i.e. the unitary operators on Hilbert spaces. In this paper we propose a generalization of Schur functions by extending the above operator representation to arbitrary closed operators on Banach spaces. Such generalized ''Schur functions'' meet the formal structure of first return generating functions, thus we call them FR-functions. We derive some general properties of FR-functions, among them a simple relation with an operator version of Stieltjes functions which generalizes the renewal equation already known for random and quantum walks. We also prove that FR-functions satisfy splitting properties which extend useful factorizations of Schur functions. When specialized to self-adjoint operators on Hilbert spaces, we show that FR-functions become Nevanlinna functions. This allows us to obtain properties of Nevanlinna functions which, as far as we know, seem to be new. The FR-function structure leads to a new operator representation of Nevanlinna functions in terms of self-adjoint operators, whose spectral measures provide also new integral representations of such functions. This allows us to characterize each Nevanlinna function by a measure on the real line, which we refer to as ''the measure of the Nevanlinna function''. In contrast to standard operator and integral representations of Nevanlinna functions, these new ones are exact analogues of those already known for Schur functions. The above results are also the source of a very simple ''Schur algorithm'' for Nevanlinna functions based on interpolations at points on the real line, which we refer to as the ''Schur algorithm on the real line''. The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions, clarifying the meaning of these functions as first return generating functions and their splittings which become recurrence splitting rules. They also show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature. (C) 2017 Elsevier Inc. All rights reserved