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SUMMARY:Sean Eberhard (Warwick)
DTSTART:20251104T140000Z
DTEND:20251104T160000Z
DTSTAMP:20260423T041614Z
UID:WienGAGT/49
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WienGAGT/49/
 ">Growth gap of residually soluble groups</a>\nby Sean Eberhard (Warwick) 
 as part of Vienna Geometry and Analysis on Groups Seminar\n\n\nAbstract\nI
 f \\(G = \\langle X\\rangle\\) is a finitely generated group\, the growth 
 function \\(\\gamma_G(n)\\) is the number of elements of \\(G\\) word leng
 th at most \\(n\\). We ignore the fine-scale detail of this function and f
 ocus on how fast it tends to infinity. The growth of a group is a fundamen
 tal quasi-isometric invariant\, but there are still many mysteries. It can
  be as slow as polynomial (e.g.\, for nilpotent groups)\, or as fast as ex
 ponential (e.g.\, for free groups)\, and nothing else was known until the 
 80's when Grigorchuk gave his famous example of a group of intermediate gr
 owth\, i.e.\, neither polynomial nor exponential\, and it is now known (Er
 schler--Zheng\, 2020) that this group has growth roughly \\(\\exp(n^{0.767
 })\\). Grigorchuk's "gap conjecture" predicts that there is some constant 
 \\(c > 0\\) such if the growth is slower than \\(\\exp(n^c)\\) then it sho
 uld be polynomial (which is equivalent to virtual nilpotence\, by a theore
 m of a Gromov). This is known for residually nilpotent groups with \\(c = 
 1/2\\)\, and Wilson (2011) showed that it holds for residually soluble gro
 ups with \\(c = 1/6\\). Elena Maini and I have now improved this to \\(c =
  1/4\\) in the residually soluble case. To be precise\, if \\(G\\) is resi
 dually soluble and its growth is \\(< \\exp(\\frac{n^{1/4}}{100})\\) for l
 arge \\(n\\) then \\(G\\) is in fact virtually nilpotent. In this talk I w
 ill give an overview of this landsacpe\, including a basic introduction to
  the theory of growth\, and by the end of the talk I will give the whole p
 roof of a slightly weaker bound with exponent \\(\\frac{1}{4.16}\\).\n
LOCATION:https://researchseminars.org/talk/WienGAGT/49/
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