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\begin{document}
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\hfil {\bf\LARGE Groups and Mosaics}
\bigskip
\mysection{Mistakes}
It is reasonable to fix some mistakes in the problems text.
\task{A4} Let us consider a plane divided into {\tt convex} {\rm7}-gons which
diameters are less or equal to $1$. Fix a point $O$. Let $N(R)$ be the number
of {\rm7}-gons falling into the circle with diameter $R$ and center $O$. Prove
that there exists $\lambda>1$ such that $N(R)> {\lambda}^R$.
\task{B1} Let $ba=ab$ be a defining relation. Prove that any word can be
transformed to the form $a^mb^n$; here $m$, $n$ are integers.
\mysection{Additional problems}
\task{A5} Can one cut a plane to convex {\rm7}-gons such that diameters of the
{\rm7}-gons are less or equal to 1 and any unite circle intersects with less
then million of them?
\task{B7.5a} Consider a group over an alphabet $\{a,\, b \}$. Prove the following
statement: if for any $x$ from a group $x^3=1$, then the group is finite.
\task{B7.5b} Consider a group over an alphabet $\{a,\, b, \, c \}$. Prove the following
statement: if for any $x$ from a group $x^3=1$, then the group is finite.
\mysection{Some additional technics}
Now we shall give the following notation. Consider a cutting of a plane to some polygons (cells).
This cutting is called a {\it map} and is denoted by $U$. An oriented edge of a cutting $U$ is
called an {\it edge of a map}. So if there exist an edge $e$, then there exist an
opposite oriented edge $e^{-1}$. This edge $e^{-1}$ belongs the same points of the plane as
the $e$ one. Let us make an agreement that outlines of all cells should be read
по часовой стрелке. Consider the outline of all map. This outline belongs $n$ edges.
Let us denote these edges $e_1$, $e_2$, \dots $e_n$ using the orientation of the map.
A lap $e_1$\dots $e_n$ is called an {\it outline of a map} $U$. We can define the
outline of a cell by the same way. We shall consider an outline of a map regardless of
cyclic shift. Suppose an outline (of a cell, or of a map) $e_1$\dots $e_n$ belongs an edge $e$.
A chain of the edges $e_1$, $e_2$, \dots $e_n$ is called a {\it way} if the end of $e_i$
is congruent to the begin of $e_{i+1}$ for any $i=1,\dots, n-1$.
A subpath is similar to a subword: $p$ is a {\it subway} of $g$ if $q=p_1p_2$ holds for
some ways $p_1$ and $p_2$. Consider a finite alphabet $L$. We denote
$\bar{L}=L\cup L^{-1} \cup 1$, here $L^{-1}$ is an alphabet of inverse letters.
Let us assign a letter $\phi (e)$ from $\bar{L}$ to each edge $e$ of map {\bf
$U$}. The map {\bf $U$} is called a {\it diagram over $U$} if $\phi
(e^{-1})=\phi (e)^{-1}$. Consider a path $p=e_1\dots e_n$ in the diagram {\bf
$U$} over $L$. A {\it label} $\phi (p)$ is a word $\phi (e_1)\dots \phi (e_n)$
in the alphabet $\bar{L}$. By definition, put $\phi (p)=1$ if $n=|p|=0$. It is
easy to see that a label of cell (or diagram) outline is defined up to cyclical
shift. So it is a cyclical word.
A cell $K$ is called a {\it $R$-cell} if it's outline label $\phi (p)$ is
graphically equal (up to cyclical shift) to some word from defining relations
or its inverse (up to pasting some amount of symbols $1$).
It is clear that if we choose the beginning and direction of ``reading'' and ignore
the symbol $1$, then we can read a defining relation word.
A cell $K$ is called a {\it 0-cell} if it's outline label $e_1 \dots e_n$ is
graphically equal $\phi (e_1)\dots \phi (e_n)$, where all $\phi (e_i)=1$ ($=$
means a graphical equality) OR if for some $i\ne j$ $\, \phi (e_i)=a$, $\phi
(e_j)=a^{-1}$ $a$ belongs to alphabet and for all other $k\ne i, \, j \,$
$ \phi (e_k)=1$. An edge is called a {\it $0$-edge} if it's label is
equal to $1$. An edge is called a {\it $U$-edge} if it's label is non trivial
word. {\it A length $|p|$} of an arbitrary path is a number of it's $U$-edges.
{\it Perimeter} of a cell or a diagram is just a length of it's outline.
There are no 0-cells in the examples presented above. However, it is
reasonable to introduce them by following reason. The examples shown on the
figures 1--3 are truly disk diagrams: if we delete the outline, then the
diagram doesn't brake onto several parts. Diagram on the picture 4 is not a
disk: if we delete the outline, then it divides into several parts. This
problem cause some technical difficulties. For example, we need to cut the
subdiagram for induction reason. If we use 0-cells, then we can make a disk
diagram with the diagram on the picture 4. It is reasonable to think of
0-cells as ``thin'' cells (or ``fat'' edges) and 0-edges as extremely
``short'' edges.
$$
\vcenter{\centering\includegraphics{earth.2}\par
\risunok\label{ris:disk}\par
}$$
$$
\vcenter{\centering\includegraphics[scale=1]{earth.9}\par
\risunok\label{ris:path}\par
}$$
Thus, a diagram over an alphabet $U$ is called a {\it diagram over group} $G$
defined by the relations $R_1$ \dots $R_n$ if it's every cell is $R$-cell or
$0$-cell.
It is reasonable to do a {\it $0$-cutting} of a diagram. Let a cell be a
polygon. We draw a second polygon inside the first one. The second polygon is
similar to the first one. Let us connect corresponding vertexes of the polygons
with additional edges. Then we label the second polygon with letters as the
first one. The additional edges are $0$-edges. Let us label them with $1$'s. So
we obtain the $0$-cutting of a cell. Similarly, we can obtain a {\it doubling}
of a path. Draw an additional edge for every edge of a path. Then we label the
additional edges by the same way as the corresponding original edges. So we
obtain two paths with the same beginning and ending.
\end{document}