{"id":47,"date":"2016-10-08T01:23:24","date_gmt":"2016-10-08T01:23:24","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=47"},"modified":"2016-10-08T01:23:24","modified_gmt":"2016-10-08T01:23:24","slug":"a-free-semigroup","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2016\/10\/08\/a-free-semigroup\/","title":{"rendered":"A free semigroup"},"content":{"rendered":"

In the appendix here<\/a>, we make the claim that the semigroup<\/p>\n

\n\"\Gamma_A=\langle\n<\/p>\n

is free on the given generators.<\/p>\n

Here, I will give a proof. It is a simpler version of the ping-pong argument<\/a> commonly used to prove that a group is free on a given set of generators.<\/p>\n

First a simple observation. it suffices to prove that the set<\/p>\n

\n\"\left\{\n<\/p>\n

generates a free semigroup.<\/p>\n

Now, consider the action of our semigroup on the interval \"(1,\infty)\" by fractional linear transformations:<\/p>\n

\n\"\begin{pmatrix}a\n<\/p>\n

In particular, note that<\/p>\n

\n\"\begin{pmatrix}a\n<\/p>\n

Now suppose that<\/p>\n

\n\"\begin{pmatrix}y_1\n<\/p>\n

Apply both sides to some \"x\in(1,\infty)\". The LHS lies in \"(y_1,y_1+1)\" and the RHS lies in \"(z_1,z_1+1)\". Therefore \"y_1=z_1\" and the rest is an easy induction.<\/p>\n","protected":false},"excerpt":{"rendered":"

In the appendix here, we make the claim that the semigroup is free on the given generators. Here, I will give a proof. It is a simpler version of the ping-pong argument commonly used to prove that a group is … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[6],"tags":[],"class_list":["post-47","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":false,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-L","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=47"}],"version-history":[{"count":10,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/47\/revisions"}],"predecessor-version":[{"id":57,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/47\/revisions\/57"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=47"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=47"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}