Algebraic Techniques for Calculating the Nielsen Number on Hyperbolic Surfaces

Hart, Evelyn L.

doi:10.1007/1-4020-3222-6_13

Cited by 13 publications

(20 citation statements)

References 21 publications

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“…Next, for a class of maps f that is slightly more general than that considered here, Wagner in [21] introduced techniques from combinatorial group theory into the game, to produce an algorithm for determining N (f ). This was continued in [22], in the work of the first author in [5], [6], and by Kim in [17].…”

mentioning

confidence: 83%

“…The usual correspondence between the geometry and the algebra is denoted by Remark 2.1 (Warning about notation). For the Reidemeister action we use the notation of the joint work of the last two authors (in [10]), which is different from, but equivalent to, the notation of [5], [21], and [8]. The Reidemeister equivalence for f n * is given by α ∼ β if and only if there exists a δ ∈ π 1 (X, x 0 ), with α = δβf n * (δ −1 ).…”

Section: Preliminariesmentioning

confidence: 99%

“…W x , W x , maps with remnant, and Wagner's algorithm. In [21], Wagner presents a method (Theorem 2.12 below) for finding Reidemeister equivalences for maps on surfaces with boundary (see also [5]). Wagner proves that for maps with a certain property (having remnant) her method is algorithmic in that it finds all Reidemeister equivalences that are needed to determine the Nielsen number.…”

Section: 2mentioning

confidence: 99%

“…A consequence of the notational choice mentioned in Remark 2.1 is that the W 's and W 's here are the inverses of those in [21] and [5]. Note in particular, as explained below, that f * (a) = W −1…”

Section: 2mentioning

confidence: 99%

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Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles I

2008

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Abstract. In this paper and its sequel we present a method that, under loose restrictions, is algorithmic for calculating the Nielsen type numbers N Φn(f ) and N Pn(f ) of self maps f of hyperbolic surfaces with boundary and also of bouquets of circles. Because self maps of these surfaces have the same homotopy type as maps on wedges of circles, and the Nielsen periodic numbers are homotopy type invariant, we need concentrate only on the latter spaces. Of course the results will then automatically apply to the former spaces as well. The algorithm requires only that f has minimal remnant, by which we mean that there is limited cancellation between the f * images of generators of the fundamental group. These methods often work even when the minimal remnant condition is not satisfied.Our methodology involves three separate techniques. Firstly, beginning with an endomorphism h on the fundamental group, we adapt an algorithm of Wagner to our setting, allowing us to distinguish non-empty Reidemeister classes for iterates of a special representative map for h, which we introduce. Secondly, using techniques reminiscent of symbolic dynamics, we assign key algebraic information to the actual periodic points of this special representative. Finally, we use word length arguments to prove that the remaining information required for the calculation of N Φn(f ) and N Pn(f ) can be found with a finite computer search. We include many illustrative examples.In this first paper we give the tools we need in order to present and give the algorithm for N Pn(f ). All the tools introduced here will be needed in the sequel where we develop the extra tools needed in order to compute N Φn(f ).

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mentioning

confidence: 83%

Section: Preliminariesmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles I

2008

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“…Simple and practical formulas have only been obtained in specific cases (for an overview, see for instance [2,3]) and one often turns the attention to comparing the Nielsen number to other numbers that are relatively more easy to compute, such as the Lefschetz number and the Reidemeister number (see for instance [4] for an overview).…”

Section: Introductionmentioning

confidence: 99%

Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds

Lee

Penninckx

2012

Fixed Point Theory Appl

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We prove practical formulas for the Reidemeister coincidence number, the Lefschetz coincidence number and the Nielsen coincidence number of continuous maps between oriented infra-nilmanifolds of equal dimension. In order to obtain these formulas, we use the averaging formulas for the Lefschetz coincidence number and for the Nielsen coincidence number and we develop an averaging formula for the Reidemeister coincidence number. We also give a simple proof of the averaging formula for the Lefschetz coincidence number. Mathematics Subject Classification 2000: 55M20; 57S30.

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The Nielsen number for free fundamental groups and maps without remnant

Hart

Kim

2007

J. fixed point theory appl.

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For Y any space that has the homotopy type of a wedge of finitely many circles, and for g : Y → Y a map, the Nielsen number of g, N (g), is a homotopy invariant lower bound for the size of the fixed point set of any map homotopic to g. Such a map g has k-remnant if, roughly, there is limited cancellation in any product g (u)g (v) where g is the induced homomorphism and u, v ∈ π1(Y ) and |u| = |v| = k. We prove that such maps are (k + 1)-characteristic, meaning that in order to determine the Nielsen classes of fixed points, we need only test whether a limited, specified, set of elements z ∈ π1(Y ) are solutions to the equation z = W −1x f (z)Wy, with x and y fixed points that are represented in the fundamental group by Wx and Wy, respectively. The number of elements to be tested is profoundly decreased by using abelianization as well.This work is a significant extension of Wagner's results involving maps with remnant and Wagner's algorithm. Our proofs involve new concepts and techniques.We present an algorithm for N (g) for any map g with k-remnant, and we provide examples for which no algebraic techniques previously known would work. One example shows that for any k there is a map that does not have (k − 1)-remnant but does have k-remnant. (2000). 55M20. Mathematics Subject Classification

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Algebraic Techniques for Calculating the Nielsen Number on Hyperbolic Surfaces

Cited by 13 publications

References 21 publications

Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles I

Algorithms for Nielsen type periodic numbers of maps with remnant on surfaces with boundary and on bouquets of circles I

Formulas for the Reidemeister, Lefschetz and Nielsen coincidence number of maps between infra-nilmanifolds

The Nielsen number for free fundamental groups and maps without remnant

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