A REFINEMENT OF THE SIMPLE CONNECTIVITY AT INFINITY OF GROUPS

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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

Sujets

• Linear
• Vietoris?Rips complex
• Sufficiently large

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: 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 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, d X )and(Y, d Y )are quasi-isometric if there are constants,Cand mapsf:X→Y,g:Y→X(called(, C)-quasi- isometries) such that the following: d Y (f(x 1 ), f(x 2 ))6d X (x 1 , x 2 ) +C,

d X (g(y 1 ), g(y 2 ))6d Y (y 1 , y 2 ) +C, d X (f g(x), x)6C, d Y (gf(y), y)6C, hold true for allx, x 1 , x 2 ∈X, y, y 1 , y 2 ∈Y. Denition2.A connected, locally compact, topological spaceXwith 1 X= 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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