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SUMMARY:Lvzhou Chen (Purdue)
DTSTART:20230516T130000Z
DTEND:20230516T150000Z
DTSTAMP:20260423T023012Z
UID:WienGAGT/35
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WienGAGT/35/
 ">The Kervaire conjecture and the minimal complexity of surfaces</a>\nby L
 vzhou Chen (Purdue) as part of Vienna Geometry and Analysis on Groups Semi
 nar\n\n\nAbstract\n<p>Talk 1 </p>\n<p>Title: Weights of groups </p>\n<p>Ab
 stract: This is an introductory talk on weights of groups. The weight (als
 o called the normal rank) of a group \\(G\\) is the smallest number of ele
 ments that normally generate \\(G\\). We will discuss basic properties and
  examples in connection to topology. Although it is a simple notion\, seve
 ral basic problems remain open\, including the Kervaire conjecture and the
  Wiegold question. We will explain some well-known partial results and the
 ir proofs. </p>\n<p>&nbsp\;</p>\n<p>Talk 2</p>\n<p>Title: The Kervaire con
 jecture and the minimal complexity of surfaces</p>\n<p>Abstract: We use to
 pological methods to solve special cases of a fundamental problem in group
  theory\, the Kervaire conjecture\, which has connection to various proble
 ms in topology. The conjecture asserts that\, for any nontrivial group \\(
 G\\) and any element \\(w\\) in the free product \\(G*Z\\)\, the quotient 
 \\((G*Z)/&lt\;&lt\;w&gt\;&gt\;\\) is still nontrivial\, i.e. the group \\(
 G*Z\\) has weight greater than 1. We interpret this as a problem of estima
 ting the minimal complexity (in terms of Euler characteristic) of surface 
 maps to certain spaces. This gives a conceptually simple proof of Klyachko
 's theorem that confirms the Kervaire conjecture for any \\(G\\) torsion-f
 ree. We also obtain injectivity of the map \\(G\\to(G*Z)/&lt\;&lt\;w&gt\;&
 gt\;\\) when \\(w\\) is a proper power for arbitrary \\(G\\). Both results
  generalize to certain HNN extensions. </p>\n<p>&nbsp\;</p>\n
LOCATION:https://researchseminars.org/talk/WienGAGT/35/
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