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diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex index 96522f5..9ec4b0c 100644 --- a/sections/conj_classes.tex +++ b/sections/conj_classes.tex @@ -32,7 +32,7 @@ We will show that the conjugacy class of $\sigma$ is an intersection of countably many comeagre sets. - Let $A_n = \{\alpha\in Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$. + Let $A_n = \{\alpha\in \Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$. This set is comeagre for every $n>0$. Indeed, we can represent this set as an intersection of countable family of open dense sets. Let $B_{n,k}$ be the set of all finite functions $\beta\colon M\to M$ that consist @@ -46,7 +46,7 @@ $\Aut(M)$: take any finite $\gamma\colon M\to M$ such that $[\gamma]_{\Aut(M)}$ is nonempty. Then also $\bigcup_{\beta\in B_{n,k}} [\beta]_{\Aut(M)}\cap[\gamma]_{\Aut(M)}\neq\emptyset$, - one can easily construct a permutation that extends $\gamma$ and have at least + one can easily construct a permutation that extends $\gamma$ and has at least $k$ many $n$-cycles. Now we see that $A = \bigcap_{n=1}^{\infty} A_n$ is a comeagre set consisting @@ -82,7 +82,7 @@ \begin{theorem} \label{theorem:generic_aut_general} - Let $\cC$ be a Fraïssé class of finite $L$-structures. + Let $\cC$ be a Fraïssé class of finitely generated $L$-structures. Let $\cD$ be the class of structures from $\cC$ with additional unary function symbol interpreted as an automorphism of the structure. If $\cC$ has the weak Hrushovski property @@ -90,14 +90,28 @@ automorphism group of the $\Flim(\cC)$. \end{theorem} + Before we get to the proof, let us establish some notions. If + $g\colon\Gamma\to\Gamma$ is a finite injective function, then we say that + $g$ is \emph{good} if it gives (in a natural way) an isomorphism between + $\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures + generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our + case, $g$ is good + if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity + of $\Gamma$). + + Also it is important to mention that an isomorphism between two finitely + generated structures is uniquely given by a map from generators of one structure + to the other. This allow us to treat a finite function as an isomorphism + of finitely generated structures. + \begin{proof} - Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. Let $G = \Aut(\Gamma)$, - i.e. $G$ is the automorphism group of $\Gamma$. First, by the Theorem + Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem \ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality - that $\Pi = \Gamma$. - We will construct a strategy for the second player in the Banach-Mazur game - on the topological space $G$. This strategy will give us a subset - $A\subseteq G$ and as we will see a subset of the $\sigma$'s conjugacy class. + that $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$, + i.e. $G$ is the automorphism group of $\Gamma$. + We will construct a winning strategy for the second player in the Banach-Mazur game + (see \ref{definition:banach-mazur-game}) + on the topological space $G$ with $A$ being $\sigma$'s conjugacy class. By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove that this class is comeagre. @@ -105,9 +119,9 @@ sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some finite set $A\subseteq \Gamma$ and some automorphism $g_0\in G$. In other words, a basic open set is a set of all extensions of some finite partial - isomorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic - open subset given by a finite partial isomorphism $g$. Note that $B_g$ - is nonemty because of ultrahomogeneity of $\Gamma$. + automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic + open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$ + is nonempty because of ultrahomogeneity of $\Gamma$. With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider only games where both players choose finite partial isomorphisms. Namely, @@ -115,15 +129,6 @@ chooses $g_0, g_1,\ldots$ such that $f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify the corresponding basic open subsets $B_{f_0}\supseteq B_{g_0}\supseteq\ldots$. - - % Our goal is to choose $g_i$ in such a manner that the resulting function - % $g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the Fraïssé limit - % $\Gamma$ such that $(\Gamma, g) = \Flim{\cD}$. - % Precisely, we will find $g_i$'s such that - % $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and by - % the Fraïssé theorem \ref{theorem:fraisse_thm} - % it will follow that $(\Gamma, g)\cong (\Pi, \sigma)$. Hence, - % by the fact \ref{fact:conjugacy}, $g$ and $\sigma$ conjugate in $G$. Our goal is to choose $g_i$ in such a manner that $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and $(\Gamma, g)$ is ultrahomogeneous @@ -131,39 +136,41 @@ that $(\Gamma, \sigma)\cong (\Gamma, g)$, thus by the Fact \ref{fact:conjugacy} we have that $\sigma$ and $g$ conjugate. - % Once again, by the Fraïssé theorem and by Lemma - % \ref{lemma:weak_ultrahom} constructing $g_i$'s in a way such that - % age of $(\Gamma, g)$ is exactly $\cD$ and so that it is weakly ultrahomogeneous - % will produce expected result. - First, let us enumerate all pairs of structures - $\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN},\in\cD$ - such that the first element of the pair embeds by inclusion in the second, - i.e. $(A_n, \alpha_n)\subseteq (B_n, \beta_n)$. Also, it may be that - $A_n$ is an empty. We enumerate the elements of the Fraïssé limit - $\Gamma = \{v_0, v_1, \ldots\}$. - Fix a bijection $\gamma\colon\bN\times\bN\to\bN$ such that for any $n, m\in\bN$ we have $\gamma(n, m) \ge n$. This bijection naturally induces a well ordering on $\bN\times\bN$. This will prove useful later, as the main ingredient of the proof will be a bookkeeping argument. For technical reasons, let $g_{-1} = \emptyset$ and - $X_{-1} = \emptyset$. + $X_{-1} = \emptyset$. Enumerate the elements of the Fraïssé limit + $\Gamma = \{v_0, v_1, \ldots\}$. Suppose that player \textit{I} in the $n$-th move chooses a finite partial - isomorphism $f_n$. We will construct a finite partial isomorphism $g_n\supseteq f_n$ - and a set $X_n\subseteq\bN^2$ - such that following properties hold: + automorphism $f_n$. We will construct a finite partial automorphism + $g_n\supseteq f_n$ together with a finitely generated substructure + $\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ + such that the following properties hold: \begin{enumerate}[label=(\roman*)] - \item $g_n$ is an automorphism of the induced substructure $\Gamma_n$, + \item $g_n$ is good and + $\dom(g_n)\cup\rng(g_n)\subseteq\langle\dom(g_n)\rangle = \Gamma_n$, + i.e. $g_n$ gives an automorphism of a finitely generated + substructure $\Gamma_n$ \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined, - \item let - $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$ - be the enumeration of all pairs of finite structures of $T$ with automorphism - such that the first is a substructure of the second, i.e. - $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$ - is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $\FrGr_{n-1}$ (which - is the substructure induced by $g_{n-1}$). Let + + \end{enumerate} + Before we give the third point, suppose recursively that $g_{n-1}$ already + satisfy all those properties. Let us enumerate + $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$ + all pairs of finitely generated structures with automorphisms such + that the first substructure embed into the second by inclusion, i.e. + $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$ + is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $(\FrGr_{n-1}, g_{n-1})$. + We allow $A_{n,k}$ to be empty. Although $g_{n-1}$ is a finite function, + we may treat it as an automorphism as we have said before. + + \begin{enumerate}[resume, label=(\roman*)] + \item + Let $(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (with the order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and $(B_{n,k}, \beta_{n,k})$ embeds in $(\FrGr_n, g_n)$ so that this diagram @@ -178,58 +185,53 @@ \end{center} \end{enumerate} - First item makes sure that no infinite orbit will be present in $g$. The - second item together with the first one are necessary for $g$ to be an - automorphism of $\Gamma$. The third item is the one that gives weak - ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed. - - First, we will suffice the item (iii). Namely, we will construct $\Gamma'_n, g'_n$ - such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of + First, we will satisfy the item (iii). Namely, we will construct $\Gamma'_n, g'_n$ + such that $g_{n-1}\subseteq g'_n$, $\Gamma_{n-1}\subseteq\Gamma'_n$, + $g'_n$ gives an automorphism of $\Gamma'_n$ + and $f_{i,j}$ extends to an embedding of $(B_{i,j}, \beta_{i,j})$ to $(\Gamma'_n, g'_n)$. But this can be easily - done by the fact, that $\cD$ has the amalgamation property. Moreover, without - the loss of generality we can assume that all embeddings are inclusions. - - \begin{center} - \begin{tikzcd} - & (\Gamma'_n, g'_n) & \\ - (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] & & (\Gamma_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\ - & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "\subseteq"'] \arrow[ul, "\subseteq"] - \end{tikzcd} - \end{center} - - By the weak ultrahomogeneity we may assume that $\Gamma'_n\subseteq \Gamma$: - - \begin{center} - \begin{tikzcd} - (B_{i,j}\cup\Gamma_{n-1}, \beta_{i,j}\cup g_{n-1}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\ - (\Gamma'_{n}, g'_n)\arrow[ur, dashed, "f"'] - \end{tikzcd} - \end{center} - - Now, by the WHP of $\cK$ we can extend the graph $\Gamma'_n\cup\{v_n\}$ together - with its partial isomorphism $g'_n$ to a graph $\Gamma_n$ together with its - automorphism $g_n\supseteq g'_n$ and without the loss of generality we + done by the fact, that $\cD$ has the amalgamation property. + + + It is important to note that $g'_n$ should be a finite function and once + again, as it is an automorphism of a finitely generated structure, we may + think it is simply a map from one generators of $\Gamma'_n$ to the + others. By the weak ultrahomogeneity of $\Gamma$, we may assume that + $\Gamma'_n\subseteq \Gamma$. + + Now, by the WHP of $\cC$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together + with its partial isomorphism $g'_n$ to a finitely generated structure $\Gamma_n$ + together with its + automorphism $g_n\supseteq g'_n$ and (again by weak ultrahomogeneity) + without the loss of generality we may assume that $\Gamma_n\subseteq\Gamma$. This way we've constructed $g_n$ that has all desired properties. Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism - of $\Gamma$ such that $(\Gamma, g)$ has the age $\cH$ and is weakly ultrahomogeneous. + of $\Gamma$ such that $(\Gamma, g)$ has the age $\cD$ and is weakly ultrahomogeneous. It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$ - and is a sum of increasing chain of finite automorphisms. + and is an union of an increasing chain of automorphisms of finitely generated + substructures. - Take any finite structure of $T$ with automorphism $(B, \beta)$. Then, there are + Take any $(B, \beta)\in\cD$. Then, there are $i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$. By the bookkeeping there was $n$ such that - $(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$. + $(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$. This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds - into $(\Gamma, g)$, thus it has age $\cH$. - With a similar argument we can see that $(\Gamma, g)$ is weakly ultrahomogeneous. - - By this we know that $g$ and $\sigma$ conjugate in $G$. As we stated in the - beginning of the proof, the set $A$ of possible outcomes of the game (i.e. - possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also - comeagre and $\sigma$ is a generic automorphism, as it contains a comeagre - set $A$. + into $(\Gamma, g)$. Thus, $\cD$ is a subclass of the age of $(\Gamma, g)$. + The other inclusion is obvious. Hence, the age of $(\Gamma, g)$ is $\cH$. + + It is also weakly ultrahomogeneous. Having $(A,\alpha)\subseteq(B,\beta)$, + and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$ + such that $(i,j) = \min\{\{0,1,\ldots n-1\}\times X_{n-1}\}$ and + $(A,\alpha) = (A_{i,j},\alpha_{i,j}), (B,\beta)=(B_{i,j},\beta_{i,j})$ and + $f = f_{i,j}$. This means that there is a compatible embedding of $(B,\beta)$ into + $(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$. + + Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$. + By this we know that $g$ and $\sigma$ are conjugate in $G$, thus player \textit{II} + have a winning strategy in the Banach-Mazur game with $A=\sigma^G$, + thus $\sigma^G$ is comeagre in $G$ and $\sigma$ is a generic automorphism. \end{proof} \begin{theorem} @@ -248,7 +250,7 @@ Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with weak Hrushovski property and canonical amalgamation. Let $\cD$ be the Fraïssé class (by the Theorem \ref{theorem:key-theorem} - of the structures of $\cC$ with additional automorphism of the strucutre. + of the structures of $\cC$ with additional automorphism of the structure. Let $\Gamma = \Flim(\cC)$. \begin{proposition} |