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Diffstat (limited to 'sections/conj_classes.tex')
| -rw-r--r-- | sections/conj_classes.tex | 72 |
1 files changed, 16 insertions, 56 deletions
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex index c0120d3..9ec4b0c 100644 --- a/sections/conj_classes.tex +++ b/sections/conj_classes.tex @@ -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 @@ -97,21 +97,21 @@ 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 $\Gamma$. + 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 structure. + of finitely generated structures. \begin{proof} 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 $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$, i.e. $G$ is the automorphism group of $\Gamma$. - 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. + 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. @@ -129,15 +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 @@ -145,24 +136,14 @@ 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 automorphism $f_n$. We will construct a finite partial automorphism $g_n\supseteq f_n$ together with a finitely generated substructure @@ -204,26 +185,13 @@ \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 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. + done by the fact, that $\cD$ has the amalgamation property. - \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} 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 @@ -231,14 +199,7 @@ others. By the weak ultrahomogeneity of $\Gamma$, we may assume that $\Gamma'_n\subseteq \Gamma$. - % \begin{center} - % \begin{tikzcd} - % B_{i,j}\cup\Gamma_{n-1} \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\ - % \Gamma'_{n}\arrow[ur, dashed, "f"'] - % \end{tikzcd} - % \end{center} - - Now, by the WHP of $\cK$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together + 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) @@ -247,7 +208,7 @@ 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 an union of an increasing chain of automorphisms of finitely generated substructures. @@ -257,7 +218,8 @@ By the bookkeeping there was $n$ such that $(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$. This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds - into $(\Gamma, g)$. Hence, the age of $(\Gamma, g)$ is $\cH$. + 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 $n\in\bN$ @@ -267,11 +229,9 @@ $(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$. Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$. - By this we know that $g$ and $\sigma$ 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 $A$. + 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} |