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	<title>392C Geometric Group Theory</title>
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	<description>Residual finiteness and word-hyperbolic groups</description>
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		<title>392C Geometric Group Theory</title>
		<link>http://392c.wordpress.com</link>
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		<item>
		<title>35. Separating quasi-convex subgroups.</title>
		<link>http://392c.wordpress.com/2009/04/22/35-separating-quasi-convex-subgroups/</link>
		<comments>http://392c.wordpress.com/2009/04/22/35-separating-quasi-convex-subgroups/#comments</comments>
		<pubDate>Wed, 22 Apr 2009 17:02:38 +0000</pubDate>
		<dc:creator>pweil</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Quasiconvex subgroups]]></category>
		<category><![CDATA[Separability]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1796</guid>
		<description><![CDATA[Last time: Theorem 21 (Groves&#8211;Manning&#8211;Osin): If is hyperbolic rel then there exists a finite subset such that if then (a) is injective; (b) is hyperbolic rel . Theorem 22 (Gromov, Olshanshkii, Delzant): If is hyperbolic relative to the infinite cyclic then there is a such that for all there exists a hyperbolic such that for [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1796&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://392c.wordpress.com/2009/04/22/35-separating-quasi-convex-subgroups/feed/</wfw:commentRss>
		<slash:comments>0</slash:comments>
	
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			<media:title type="html">pweil</media:title>
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		<item>
		<title>34. Combinatorial Dehn Filling</title>
		<link>http://392c.wordpress.com/2009/04/20/34-combinatorial-dehn-filling/</link>
		<comments>http://392c.wordpress.com/2009/04/20/34-combinatorial-dehn-filling/#comments</comments>
		<pubDate>Tue, 21 Apr 2009 05:06:51 +0000</pubDate>
		<dc:creator>jmaciejewski</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Hyperbolic groups]]></category>
		<category><![CDATA[Relatively hyperbolic groups]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1734</guid>
		<description><![CDATA[Recall, that for any graph we built a combinatorial horoball .  For a group and a collection of subgroups and a generating set , we built the augmented Cayley graph by gluing copies of .   is hyperbolic relative to if and only if is Gromov hyperbolic. Exercise 28: If and are finitely generated, then [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1734&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">jmaciejewski</media:title>
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		<item>
		<title>33. Relatively Hyperbolic Groups</title>
		<link>http://392c.wordpress.com/2009/04/19/33-relatively-hyperbolic-groups/</link>
		<comments>http://392c.wordpress.com/2009/04/19/33-relatively-hyperbolic-groups/#comments</comments>
		<pubDate>Mon, 20 Apr 2009 05:15:10 +0000</pubDate>
		<dc:creator>elandes</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Relatively hyperbolic groups]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1701</guid>
		<description><![CDATA[Some intuition: Recall that if is a closed hyperbolic manifold then is word-hyperbolic. However, a lot of interesting hyperbolic manifolds are not closed. Example: Let be the figure 8 knot. Then the complement admits a complete hyperbolic metric and is of finite volume. So, here we have an example of a hyperbolic manifold which is [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1701&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>1</slash:comments>
	
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			<media:title type="html">elandes</media:title>
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			<media:title type="html">figure 1</media:title>
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			<media:title type="html">fig2</media:title>
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			<media:title type="html">fig32</media:title>
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			<media:title type="html">fig42</media:title>
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			<media:title type="html">lecture_4_17_09_xymatrix1</media:title>
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			<media:title type="html">lecture_4_17_09_xymatrix2</media:title>
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			<media:title type="html">fig6</media:title>
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			<media:title type="html">fig7</media:title>
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		<item>
		<title>32. Grushko&#8217;s Theorem</title>
		<link>http://392c.wordpress.com/2009/04/16/32-grushkos-theorem/</link>
		<comments>http://392c.wordpress.com/2009/04/16/32-grushkos-theorem/#comments</comments>
		<pubDate>Thu, 16 Apr 2009 17:26:48 +0000</pubDate>
		<dc:creator>kczar44</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Graphs of groups]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1662</guid>
		<description><![CDATA[Definition. A group splits freely if acts on a tree without global fixed point and such that every edge stailizer is trivial. If does not split freely, then is called freely indecomposable. Example. . Equivalently, acts on without global fixed points. So splits freely. If but splits freely, then for . Definition. The rank of [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1662&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://392c.wordpress.com/2009/04/16/32-grushkos-theorem/feed/</wfw:commentRss>
		<slash:comments>6</slash:comments>
	
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			<media:title type="html">kczar44</media:title>
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		<item>
		<title>31. Doubles &amp; virtual retractions</title>
		<link>http://392c.wordpress.com/2009/04/15/31-doubles-virtual-retractions/</link>
		<comments>http://392c.wordpress.com/2009/04/15/31-doubles-virtual-retractions/#comments</comments>
		<pubDate>Wed, 15 Apr 2009 08:45:35 +0000</pubDate>
		<dc:creator>smhart</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Doubles of free groups]]></category>
		<category><![CDATA[Virtual retractions]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1660</guid>
		<description><![CDATA[Last time, we used the following lemma without justification, so let&#8217;s prove it now. Lemma 30. Let be a graph of groups with finite and finitely generated. If is finitely generated for every edge , then is finitely generated for every . This is not completely trivial: it is certainly possible for finitely generated group [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1660&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://392c.wordpress.com/2009/04/15/31-doubles-virtual-retractions/feed/</wfw:commentRss>
		<slash:comments>0</slash:comments>
	
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			<media:title type="html">smhart</media:title>
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		<item>
		<title>30. Doubles of Free Groups (continued)</title>
		<link>http://392c.wordpress.com/2009/04/13/30-doubles-of-free-groups-continued/</link>
		<comments>http://392c.wordpress.com/2009/04/13/30-doubles-of-free-groups-continued/#comments</comments>
		<pubDate>Mon, 13 Apr 2009 19:35:01 +0000</pubDate>
		<dc:creator>dgirao</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Doubles of free groups]]></category>
		<category><![CDATA[Elevations]]></category>
		<category><![CDATA[Separability]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1483</guid>
		<description><![CDATA[Lemma 29: Suppose is a finite set of infinite degree elvations and is compact. Then for all sufficiently large , there exists an intermediate covering such that (a) embeds in (b) every descends to an elevation of degree (c) the are pairwise distinct Proof: We claim that the images of never share an infinite ray [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1483&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
		<wfw:commentRss>http://392c.wordpress.com/2009/04/13/30-doubles-of-free-groups-continued/feed/</wfw:commentRss>
		<slash:comments>0</slash:comments>
	
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			<media:title type="html">dgirao</media:title>
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			<media:title type="html">eq.latex</media:title>
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	</item>
		<item>
		<title>29. Doubles of free groups</title>
		<link>http://392c.wordpress.com/2009/04/08/29-doubles-of-free-groups/</link>
		<comments>http://392c.wordpress.com/2009/04/08/29-doubles-of-free-groups/#comments</comments>
		<pubDate>Thu, 09 Apr 2009 03:59:31 +0000</pubDate>
		<dc:creator>yyao392c</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Doubles of free groups]]></category>
		<category><![CDATA[Elevations]]></category>
		<category><![CDATA[Separability]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1422</guid>
		<description><![CDATA[Agol-Groves-Manning&#8217;s Theorem predicts that, for every word-hyperbolic group we can easily construct, every quasiconvex subgroup is separable (otherwise, we would find a non-residually finite hyperbolic group!). In this section, we use graphs of groups to build new hyperbolic groups: Combination Theorem (Bestvina &#38; Feighn): If is a malnormal subgroup of hyperbolic groups , then is [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1422&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">yyao392c</media:title>
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			<media:title type="html">diagram</media:title>
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			<media:title type="html">diagram1</media:title>
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		<item>
		<title>28. Separability properties of amalgams</title>
		<link>http://392c.wordpress.com/2009/04/08/separability-properties-of-amalgams/</link>
		<comments>http://392c.wordpress.com/2009/04/08/separability-properties-of-amalgams/#comments</comments>
		<pubDate>Thu, 09 Apr 2009 02:01:20 +0000</pubDate>
		<dc:creator>drosen2000</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Graphs of groups]]></category>
		<category><![CDATA[Residual finiteness]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1400</guid>
		<description><![CDATA[As before, are and are graphs of spaces equipped with maps , , , and such that commutes. Lemma 28: Suppose that every edge map of is an elevation.  Then the map is -injective. Proof: The idea is to add extra vertex spaces to so that satisfies Stalling&#8217;s condition.  As before, we have inclusions: If [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1400&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">drosen2000</media:title>
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			<media:title type="html">fig_28_11</media:title>
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			<media:title type="html">fig_28_2</media:title>
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			<media:title type="html">fig_28_31</media:title>
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			<media:title type="html">fig_28_81</media:title>
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		<item>
		<title>27. Stallings&#8217; Condition</title>
		<link>http://392c.wordpress.com/2009/04/06/27-stallings-condition/</link>
		<comments>http://392c.wordpress.com/2009/04/06/27-stallings-condition/#comments</comments>
		<pubDate>Mon, 06 Apr 2009 13:44:57 +0000</pubDate>
		<dc:creator>mathjoker</dc:creator>
				<category><![CDATA[Course notes]]></category>
		<category><![CDATA[Elevations]]></category>
		<category><![CDATA[Graphs of groups]]></category>
		<category><![CDATA[Separability]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1344</guid>
		<description><![CDATA[Recall that our goal is to determine, for a given map of graphs of spaces such as the one shown below, whether the map can be extended to a covering map . Let be graphs of spaces equipped with maps , and as before.  Recall that in Stallings&#8217; proof of Hall&#8217;s Theorem, we completed an [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1344&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>1</slash:comments>
	
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			<media:title type="html">mathjoker</media:title>
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			<media:title type="html">elevations1</media:title>
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			<media:title type="html">diagram1</media:title>
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			<media:title type="html">diagram1</media:title>
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			<media:title type="html">diagram2</media:title>
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			<media:title type="html">diagram3</media:title>
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			<media:title type="html">diagram4</media:title>
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			<media:title type="html">diagram5</media:title>
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		<item>
		<title>26. Building Covering Spaces</title>
		<link>http://392c.wordpress.com/2009/04/06/26-building-covering-spaces/</link>
		<comments>http://392c.wordpress.com/2009/04/06/26-building-covering-spaces/#comments</comments>
		<pubDate>Mon, 06 Apr 2009 06:03:02 +0000</pubDate>
		<dc:creator>krammai</dc:creator>
				<category><![CDATA[Course notes]]></category>

		<guid isPermaLink="false">http://392c.wordpress.com/?p=1373</guid>
		<description><![CDATA[Lemma 27 Revisited. Suppose is a covering map. Then there is a covering map such that is the fibre product of and . Proof. Let be the fibre product of and . There is a map given by .  Let be the fibre product of and ; i.e. . There is a map given by [...]<img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=392c.wordpress.com&#038;blog=5923208&#038;post=1373&#038;subd=392c&#038;ref=&#038;feed=1" width="1" height="1" />]]></description>
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		<slash:comments>0</slash:comments>
	
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			<media:title type="html">krammai</media:title>
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