%Strg c t p fuer pdflatex \documentclass[12pt,german,xcolor=dvipsnames]{beamer} \usepackage{graphicx} \usepackage{amsmath} \usepackage{rotating} \usepackage[german]{babel} \usepackage{enumerate} \usepackage{xspace} %\usepackage{showkeys} \usepackage{young} \usepackage{makeidx} \usepackage{newlfont} \usepackage{amssymb} \usepackage{amsxtra} \usepackage{amsfonts} \usepackage{amscd} \usepackage{mathrsfs} \usepackage{stmaryrd} \usepackage{psfrag} \usepackage{color} \usepackage[latin1]{inputenc} \usepackage{times} \usepackage[mathscr]{euscript} \usepackage[all]{xy} \usepackage[active]{srcltx} \input epsf \DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} \setlength{\arraycolsep}{2pt} \input{newcommands} \newcommand{\wt}{\tilde} \newcommand{\wh}{\hat} \newcommand{\td}{\tilde} \newcommand{\eps}{\varepsilon} \title[]{~\\[2ex] Euklid's plane through Symmetry} \author[]{Wolfgang Soergel} \institute[]{\inst{} Mathematisches Institut\\ Universit\"at Freiburg\\[4ex] \vspace*{.9cm} %\textcolor{red}{\\ allgemeine Fragen/Hinweise hier positionieren} } \date[]{\small \hbox{October 2022}} \beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}} % ================================================== % ================================================== \begin{document} % ================================================== % ================================================== \titlepage \begin{frame} \begin{itemize} \item{\bf Incidence geometry:} Pair $(X,G)$ with $X$ a set of ``points'', $G\subset \mathcal P(X)$ a set of ``lines'', each line has at least two points, through every two distinct points there goes exactly one line.%\pause \item{\bf Betweeness:} Subset $Z\subset X^3$ of collinear tripels giving two opposite orders on every line, such that a line never meets only one segment of a triangle.%\pause \item{\color{Red}\bf Congruence group:} A subgroup $K\subset \op{Aut}(X,G,Z)$ such that for any two halflines $A,B\subset X$ there exist exactly two $k,h\in K$ with $k(A)=B=h(A)$.\\[4mm]%\pause \item{\bf Supremum property:} With respect to a $Z$-order every nonempty bounded above subset of a line has a supremum.%\pause \item{\bf Parallel Axiom:} $\forall g\in G, p\in X\backslash g$ there exists uniquely $ h\in G$ with $p\in h$ and $h\cap g=\emptyset$.%\pause \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item{\bf Theorem:} There is up to isomorphism a unique quadrupel $(X,G,Z,K)$ of an incidence geometry with betweeness relation and congruence group that satisfies supremum and parallel axioms and has at least one line. {{\color{Blue} \href{http://home.mathematik.uni-freiburg.de/soergel/Skripten/XXEL.pdf}{[Soergel: Elementargeometrie]}}} \\[4mm]%\pause \item A {\bf Congruence group} $K\subset \op{Aut}(X,G,Z)$ is a subgroup such that for any two halflines $A,B\subset X$ exist exactly two $k,h\in K$ with $k(A)=B=h(A)$.%\pause \item If we ask instead congruences to act free and transitive on the set of halflines, there are is a ``bad'' model for every nontrivial group homomorphism $\op{SO}(2)\ra \DR_{>0}$.%\pause \item If we ask the parallel axiom to be false, there should be also a unique model. I would like an easy proof. \end{itemize} \end{frame} \begin{frame} Now I am going to make lots of claims and if you don't believe one of them, you are welcome to speak up and I will try to explain the proof on the blackboard. \end{frame} \begin{frame} Let $(X,G,Z)$ be an incidence geometry with betweeness. \begin{itemize} \item A line meeting no vertex of a triangle meets exactly to segments or none.%\pause \item The complement of a line is the disjoint union of at most two equivalence classes under the relation ``joinable by a segment''. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be an incidence geometry with betweeness and congruences. \begin{itemize} \item Every halfline is infinite.%\pause \item For every line $g$ there is a unique nontrivial congruence $s_g$ fixing it pointwise, the {\bf reflection along $g$}.%\pause \item Every segment is infinite.%\pause \item For every line $g$ there are exactly two halfspaces. They are exchanged by the reflection $s_g$. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be an incidence geometry with betweeness and congruences. \begin{itemize} \item Let $h\perp g$ mean $g\neq h=s_g(h)$. %\pause \item For every line $g$ and every point $x$ there is a unique perpendicular $h\perp g$ with $x\in h$.%\pause \item $h\perp g\IFF s_hs_g=s_gs_h$%\pause \item Two perpendiculars to a line $g$ are disjoint.%\pause \item Under the parallel axiom perpendiculars to a line are perpendicular to its parallels. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be incidence geometry with betweeness and congruences and let $g\in G$ a line. \begin{itemize} \item Denote by $K_{|g}\subset K$ the stabilizer of a line and its halfspaces. %\pause \item Denote by $\vec g\subset K_{|g}$ the subgroup stabilizing both $Z$-orders on the line $g$.%\pause \item $\vec g$ acts free and transitive on $g$. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be an incidence geometry with betweeness and congruences and supremum axiom and let $g\in G$ be a line. \begin{itemize} \item Given $v\in\vec g\backslash e_K$ we have $v(x)>x$ for all $x$ and some $Z$-order on $g$.%\pause \item All elements of $K_{|g}\backslash \vec g$ are involutions.%\pause \item Any two different points $x\neq y$ can be exchanged by a unique reflection. It stabilizes the $\overline{xy}$-halfplanes. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be an incidence geometry with betweeness and congruences and supremum axiom and let $g\in G$ be a line. \begin{itemize} \item All elements of $K_{|g}\backslash \vec g$ are reflections. These elements generate $K_{|g}$.%\pause \item Every element $v\in \vec g$ has a square root. Conjugating $v$ by an element of $K_{|g}\backslash \vec g$ we get its inverse.%\pause \item $\vec g$ is commutative. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be an incidence geometry with betweeness and congruences and supremum and parallels axiom. \begin{itemize} \item $g\parallel h\RA \vec g=\vec h$ %\pause \item Assume there exists a line. Then all translations form a commutative subgroup $\vec X\pdef \bigcup_{g\in G}\vec g\subset K$ acting free and transitive on $X$. \end{itemize} \end{frame} \begin{frame} \begin{itemize} \item Let $A$ be an ordered group such that no nontrivial cyclic subgroup has an upper bound in $A$ and every element has a root. Then for every $a>e_A$ there exists a unique order preserving group homomorphism $A\ra (\DR,+)$ with $a\mapsto 1$ and it is injective. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be incidence geometry with betweeness and congruences and supremum axiom. \begin{itemize} \item For every line $g$, there is a unique structure on $\vec g$ as a real vector space compatible with $Z$. %\pause \item If the parallel axiom holds and there is a least one line, there is a unique structure of real vector space on $\vec X$ such that $\vec g$ is a subspace for all $g\in G$. Furthermore the space $\vec X$ has dimension two, so that $X$ acquires the structure of a twodimensional real affine space. \end{itemize} \end{frame} \begin{frame} Let $(X,G,Z,K)$ be incidence geometry with betweeness and congruences and supremum and parallel axioms and a line. \begin{itemize} \item The congruence group $K$ consists of affinities and contains all translations.%\pause \item The isotropy groups $K_x$ all have the same image $D\subset \op{GL}(\vec X)$.%\pause \item The subgroup $D\subset \op{GL}(\vec X)$ has the property that for any two rays $A,B\subset \vec X$ there are exactly two elements $r,s\in D$ with $r(A)=B=s(A)$. \end{itemize} \end{frame} \begin{frame} Let $V$ be a twodimensinal real vector space. Let $D\subset \op{GL}(V)$ a subgroup such that for any two rays $A,B\subset V$ there are exactly two elements $r,s\in D$ with $r(A)=B=s(A)$. \begin{itemize} \item There exists a $D$-invariant scalar product on $V$.%\pause \item Any two $D$-invariant scalar products are scalar multiples of one another.%\pause \item The group $D$ is the orthogonal group for one and any of these scalar products. \end{itemize} \end{frame} \begin{frame} This proves that an incidence geometry with betweeness and congruences $(X,G,Z,K)$ satisfying supremum and parallel axioms and having a line is isomorphic to $$(\DR^2, \text{affine lines}, \text{obvious } Z, \op{O}(2)_{\op{aff}})$$ \end{frame} \begin{frame} Thanks! \end{frame} \end{document} %scp /home/wolfgang/Dokumente/Skripten/Skripten/VortragDubrovnikEuk.pdf soergel@tux00.mathematik.uni-freiburg.de:/webserver/home/soergel/PReprints/VortragDubrovnikEuk.pdf %%% Local Variables: %%% mode: pdflatex %%% End: