A REFINEMENT OF THE SIMPLE CONNECTIVITY AT INFINITY OF GROUPS
8 pages
English

A REFINEMENT OF THE SIMPLE CONNECTIVITY AT INFINITY OF GROUPS

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8 pages
English
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A REFINEMENT OF THE SIMPLE CONNECTIVITY AT INFINITY OF GROUPS LOUIS FUNAR AND DANIELE ETTORE OTERA Abstract. We give another proof for a result of Brick ([2]) stating that the simple connectivity at infinity is a geometric property of finitely presented groups. This allows us to define the rate of vanishing of pi∞1 for those groups which are simply connected at infinity. Further we show that this rate is linear for cocompact lattices in nilpotent and semi-simple Lie groups, and in particular for fundamental groups of geometric 3-manifolds. Keywords: Simple connectivity at infinity, quasi-isometry, colored Rips com- plex, Lie groups, geometric 3-manifolds. MSC Subject: 20 F 32, 57 M 50. 1. Introduction The first aim of this note is to prove the quasi-isometry invariance of the simple connectivity at infinity for groups, in contrast with the case of spaces. We recall that: Definition 1. The metric spaces (X, dX) and (Y, dY ) are quasi-isometric if there are constants ?, C and maps f : X ? Y , g : Y ? X (called (?,C)-quasi- isometries) such that the following: dY (f(x1), f(x2)) 6 ?dX (x1, x2) + C, dX(g(y1), g(y2)) 6 ?dY (y1, y2) + C, dX(fg(x), x) 6 C, dY (gf(y

  • linear

  • all group

  • group

  • rips complex

  • has no

  • being adjacent

  • pi∞1

  • immediate now

  • isometry invariant

  • large enough


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A REFINEMENT OF THE SIMPLE CONNECTIVITY AT INFINITY OF GROUPS
LOUIS FUNAR AND
DANIELE ETTORE OTERA
Abstract.We give another proof for a result of Brick ([2]) stating that the simpleconnectivityatin nityisageometricpropertyof nitelypresented groups.Thisallowsustode netherateofvanishingoffor those groups 1 whicharesimplyconnectedatin nity.Furtherweshowthatthisrateis linear for cocompact lattices in nilpotent and semi-simple Lie groups, and in particular for fundamental groups of geometric 3-manifolds. Keywords:scomdRip-itivctneonecplimSlorey,cometr-isoauisytq, ninayit plex, Lie groups, geometric 3-manifolds. MSC Subject:20 F 32, 57 M 50.
1.Introduction The rstaimofthisnoteistoprovethequasi-isometryinvarianceofthesimple connectivityatin nityforgroups,incontrastwiththecaseofspaces.Werecall that:
De nition1.The metric spaces(X, dX)and(Y, dY)are quasi-isometric if there are constants,Cand mapsf:XY,g:YX(called(, C)-quasi-isometries) such that the following: dY(f(x1), f(x2))6dX(x1, x2) +C,
dX(g(y1), g(y2))6dY(y1, y2) +C, dX(f g(x), x)6C, dY(gf(y), y)6C, hold true for allx, x1, x2X, y, y1, y2Y. De nition2.A connected, locally compact, topological spaceXwith1X= 0is simplyconnectedatin nity(abbreviateds.c.i.andonewritesalso X= 0) if for 1 each compactkXthere exists a larger compactkKXsuch that any closed loop inXKis null homotopic inXk.
Remark 1.Tmplehesiitcennocnitaytivsnyiit nsiuaaqottnnvyiiaarso-itrme 1 2 1 2 of spaces ([15]fact). In (SR)Dand(SR)Dare simply 1 1 SZS{0} connected,quasi-isometricspacesalthoughthe rstissimplyconnectedatin nity while the second is not.
This notion extends to a group-theoretical framework as follows (see [3], p.216):
Partially supported by GNSAGA.
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