Fundamental groups, slalom curves and
We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from -braid invariants and their application.
Key words and phrases:fundamental group, extremal length, conformal module, -braids
1991 Mathematics Subject Classification:Primary 30Cxx; Secondary 20F34,20F36,57Mxx
In this paper we will describe a conformal invariant for the elements of the fundamental group of the twice punctured complex plane with base point and give upper and lower bounds for this invariant. The group is a free group with two generators. We choose generators and so that is represented by a simple closed curve with base point which surrounds the point counterclockwise such that the image of the curve except the point is contained in the left half-plane. Respectively, a standard representative of the generator surrounds the point counterclockwise and the image of the curve except the point is contained in the right half-plane.
The fundamental group is isomorphic to the relative fundamental group whose elements are homotopy classes of curves in with end points on the interval . We refer to as fundamental group with totally real boundary values (-boundary values for short). To establish the isomorphism one has to use that the set is connected and simply connected and contains . In the same way is isomorphic to the relative fundamental group with perpendicular bisector boundary values whose elements are homotopy classes of curves in with end points on the imaginary axis . For an element we denote by , and by , respectively, the elements in , and in , respectively, corresponding to .
Consider a rectangle with sides parallel to the axes and with length of the horizontal sides equal to b and length of the vertical sides equal to a. Recall that according to Ahlfors’s definition  the extremal length of such a rectangle is equal to and its conformal module equals . A continuous mapping of the rectangle into is said to represent if it has a continuous extension to the closure of which maps horizontal sides to the interval and whose restriction to each vertical side represents . We make the respective convention for instead of .
We are now in the position to define the extremal length of elements of the relative fundamental groups.
For an element of the fundamental group the extremal length of with perpendicular bisector boundary values (-boundary values for short) is defined as
An analogous definition can be given for .
We will give upper and lower bounds for and differing by a multiplicative constant. This is of independent interest for the fundamental group of the twice punctured plane, but the main motivation was to give estimates of conformal invariants of braids. Recall that a pure geometric -braid with base point is a continuous mapping of the unit interval into -dimensional configuration space whose values at the endpoints are equal to a given base point in . More geometrically, a pure geometric -braid consists of pairwise disjoint curves in the cylinder , each joining a point in the top of the cylinder with its copy in the bottom so that for each curve the canonical projection to the interval is a homeomorphism. A pure -braid with base point is an isotopy class of pure geometric -braids with fixed base point.
Consider a pure geometric -braid. Associate to it a curve in as follows. For a point we denote by the Möbius transformation that maps to , to and fixes . Then omits , and . Notice that is equal to the cross ratio .
Let be a curve in . Associate to it the curve in which omits the points and . If is a loop with base point then is a loop with base point . The homotopy class of in with base point depends only on the homotopy class of in the configuration space with base point . We obtain a surjective homomorphism from the fundamental group of with base point to the fundamental group of with base point . The kernel of equals , the subgroup of generated by the full twist obtained by twisting the cylinder keeping the bottom fixed and turning the top by the angle . Respective facts hold for loops in with specified boundary values instead of loops with a base point. With the natural definition of the extremal length with totally real boundary values of a pure -braid this extremal length is equal to . The respective fact holds for perpendicular bisector boundary values. The obtained invariants are invariants of -braids rather than invariants of conjugacy classes of -braids. In particular, they are finer than a popular invariant of braids, the entropy. Our estimates imply estimates of the entropy of pure -braids in terms of the representing word of the image . Notice that the name ”totally real” and ”perpendicular bisector” is motivated by the definition in the case of braids. Details will be given in a later paper. For an introduction to braids see e.g. . For more information on the conformal module, the extremal length and entropy of braids, or of conjugacy classes of braids, respectively, see also  and .
We will now lift the elements of to the logarithmic covering of and identify the lifts with homotopy slalom curves. This geometric interpretation will suggest how to estimate the extremal length with perpendicular bisector boundary values.
The logarithmic covering of is the universal covering of the twice punctured Riemann sphere with all preimages of under the covering map removed. Geometrically the universal covering of can be described as follows. Take copies of labeled by the set of integer numbers. Close up each copy by attaching two copies of , the -edge (the accumulation set of points of the upper half-plane) and the -edge (the accumulation set of points of the lower half-plane). For each we glue the -edge of the -th copy to the -edge of the -st copy (using the identity mapping on to identify points on different edges). Denote by the set obtained from the described covering by removing all preimages of .
The following proposition holds.
The set is conformally equivalent to . The mapping , , , is a covering map from to .
The lift of with initial point is a curve which joins with and is contained in the closed left half-plane. The only points on the imaginary axis are the endpoints.
The lift of with initial point is a curve which joins with and is contained in the closed right half-plane. The only points on the imaginary axis are the endpoints.
Figure 1 shows the curves and which represent the generators of the fundamental group and their lifts under the covering maps and . For the curves and are the two lifts of under the double branched covering with branch points and . The curve is the lift of under the mapping with initial point , the curve lifts and has initial point .
Consider the curve . It runs times along the curve if and times along the curve which is inverse to if . For each the curve lifts to a curve with initial point and terminating point which is contained in the closed left half-plane and omits the points in . Respectively, lifts to a curve with initial point and terminating point which is contained in the closed right half-plane and omits the points in . The mentioned lifts are homotopic through curves in with endpoints on to curves with interior contained in the open (left, respectively, right) half-plane. We have the following definition where we identify a curve with its image, ignoring orientation.
A simple arc in with endpoints on different connected components of is called an elementary slalom curve if its interior (i.e. the complement of its endpoints) is contained in one of the open half-planes or .
A curve in is called an elementary half slalom curve if one of the endpoints is contained in a horizontal line for an integer and the union of the curve with its mirror reflection in the line is an elementary slalom curve.
A slalom curve in is a curve which can be divided into a finite number of elementary slalom curves so that consecutive elementary slalom curves are contained in different half-planes.
A curve which is homotopic to a slalom curve in through curves with endpoints in is called a homotopy slalom curve.
Figure 2 below shows a slalom curve which represents a lift of the element with perpendicular bisector boundary values.
Elementary slalom curves and elementary half slalom curves will serve as building blocks. Note that each curve in with endpoints in is a homotopy slalom curve or is homotopic to the identity in with endpoints in . Proposition 1 implies that each lift of a curve in with endpoints on the imaginary axis is of such type. The extremal length of slalom curves (more precisely of homotopy classes of slalom curves) can be defined in the same way as the respective object for elements of the fundamental group . The extremal length with perpendicular bisector boundary values of an element of is equal to that of its lift.
Consider the extremal length of an elementary slalom curve which corresponds to the word where equals either or . Without loss of generality we may assume that , hence the curve is contained in the closed left half-plane. After a translation the endpoints of the curve are contained in the intervals and , respectively, with . For (thus for ) the extremal length equals . In this case we call the original curve a trivial elementary slalom curve. Let be positive. The curve is represented by the extension to the boundary of a conformal mapping of an open rectangle onto the left half-plane which maps the horizontal sides onto , and , respectively. Hence its extremal length is bounded from above by the extremal length of the rectangle .
The conformal mapping of a rectangle onto the left half-plane whose extension to the boundary maps the horizontal sides onto , and , respectively, is related to elliptic integrals. With a suitable normalization of the rectangle the inverse of the mapping is equal to the elliptic integral
We use the branch of the square root which is positive on the positive real axis. The function extends continuously to the imaginary axis (the integral converges). The extended map maps the closed left half-plane to a closed rectangle. The points , , and are mapped to the vertices of the rectangle. It is known and follows from formula (Fundamental groups, slalom curves and extremal length) for the elliptic integral that for the extremal length of the rectangle satisfies the inequalities
for positive constants and not depending on .
Equation (2) suggests the following proposition.
The extremal length of an elementary slalom curve with endpoints in the intervals and , respectively, with , satisfies the inequalities
for positive constants and not depending on and .
There are explicit estimates for the constants and .
The proof will be given elsewhere. Here we already discussed the estimate from above. The estimate from below is more subtle. The first difficulty is that the representing mappings for an elementary slalom curve are not necessarily conformal mappings, they are merely holomorphic. The second difficulty is that the image of the rectangle is not necessarily contained in the half plane. We can only say about the mapping that it lifts to a holomorphic mapping into the universal covering of with specified boundary values. The universal covering is a half-plane, but the horizontal sides of the rectangle are not mapped any more into boundary intervals of the half-plane but into some curves in the half-plane. One tool for dealing with these difficulties is an analog of the following lemma which is of independent interest.
Let and be rectangles with sides parallel to the axes. Suppose is a vertical strip bounded by the two vertical lines which are prolongations of the vertical sides of the rectangle . Let be a holomorphic map whose extension to the closure maps the two horizontal sides of into different horizontal sides of . Then
Equality holds if and only if the mapping is a surjective conformal map from to .
The proof of the lemma is based on the Cauchy-Riemann equations.
To estimate the extremal length of an arbitrary element of the fundamental group we represent the element as a word in the generators (and identify it with the word). A word is in reduced form (or a reduced word) if it is written as product of powers of generators where consecutive terms correspond to different generators. Consider the reduced word
where the are integers. (Here , we allow .) We are interested first in the extremal length with perpendicular boundary value conditions. One can show that any curve which represents this element with perpendicular boundary values can be represented as composition of curves , ,…, which represent , ,…, with perpendicular boundary values. Together with Theorems 2 and 4 of  this implies the following estimate from below
This gives a good lower bound if all terms of the reduced word enter with power of absolute value at least . It does not give a good lower bound, for example, for the word with , or for the word for integers and larger than and much bigger than . In the first example the reason is the following. Each representing curve for with -boundary values can be written as composition of the following two curves: a curve with -boundary values on the left and -boundary values on the right representing , and a curve with -boundary values on the left and -boundary values on the right representing . The lift of each of the two curves is a non-trivial half-slalom curve. Hence the extremal length with -boundary values of the element is proportional to .
For the second example one can show that each representing curve with -boundary values contains a piece corresponding to with mixed boundary values. A different choice of a lift gives a half slalom curve which shows that the extremal length of this piece is proportional to .
The discussion suggests that the extremal length of a general element of can be given in terms of a syllable decomposition of the representing reduced word.
We describe now the syllable decomposition of the word (4).
Any term of the reduced word with is a syllable.
Any maximal sequence of consecutive terms of the reduced word which have equal power equal to either or is a syllable.
Each remaining term of the reduced word is characterized by the following properties. It enters with power or and the neighbouring term on the right (if there is one) and also the neighbouring term on the left (if there is one) has power different from that of the given one. Each term of this type is a syllable, called a singleton.
Define the degree of a syllable to be the sum of the absolute values of the powers of terms entering the syllable.
For example, the syllables of the word (see Figure 2) from left to right are the singleton , the syllable of degree , the syllable of degree , the syllable of degree , the singleton and the singleton .
The following theorem holds.
There are absolute positive constants and such that the following holds. Let be the word representing an element of . Then
, except in the following cases: or for an integer . In these cases .
, except in the following case: each term in the reduced word has the same power, which equals either or . In these cases .
For an element which is not one of the exceptional cases of Theorem 1 the two versions of the extremal length are comparable:
for positive constants and which do not depend on .
There are positive constants and such that for each element which is not a singleton the estimate
The extremal length of elements of the fundamental group of the complex plane with an arbitrary number of punctures will be treated in a forthcoming paper. The case of -braids with arbitrary is more subtle.
During the work on the present paper the author was supported by the SFB ”Space-Time-Matter ” at Humboldt-University Berlin. She is grateful to A. Khrabrov and K. Mshagskiy for the professional drawing of the figures.
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