K2 of discrete valuation rings

Dennis, R. Keith; Stein, Michael R.

doi:10.1016/0001-8708(75)90157-7

Cited by 78 publications

(41 citation statements)

References 8 publications

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“…We can show that there is also a dense subgroup which has property τ . 4 In order to simplify the argument we will construct a counter example inside the product l SL 3l (F p ):…”

Section: Unbounded Rank Expandersmentioning

confidence: 99%

Universal lattices and unbounded rank expanders

Kassabov

2007

Invent. math.

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We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal latticeswith respect to several generating sets.As an application of the above result we show that the Cayley graphs of the finite groups SL 3k (Fp) can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and Benjamin Weiss.

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“…We can show that there is also a dense subgroup which has property τ . 4 In order to simplify the argument we will construct a counter example inside the product l SL 3l (F p ):…”

Section: Unbounded Rank Expandersmentioning

confidence: 99%

Universal lattices and unbounded rank expanders

Kassabov

2007

Invent. math.

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“…In their paper [4] Dennis and Stein proved several new identities for the Steinberg symbols besides the standard one we quoted: …”

Section: Lemma 33 If the Finite Local Ringr Is An Image Ofmentioning

confidence: 99%

Universal lattices and property tau

Kassabov

Nikolov

2006

Invent. math.

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We prove that the universal lattices -the groups G = SL d (R) where R = Z[x1, . . . , x k ], have property τ for d ≥ 3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ -constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K2 of some finite commutative rings and a relative property T of (SL2(R) ⋉ R 2 , R 2 ).

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“…The prime-to-p part is given by the prime-to-p part of K i (O p /p) by Suslin [21]. The groups K 2 (O p /p n ) have been calculated by Dennis and Stein [5]. In the totally ramified case, the groups K i (F q [t]/t 2 ) have been determined by Hesselholt and Madsen [10] and in the unramified case, Evens and Friedlander [7] proved K 3 (Z/p 2 ) p = Z/p 2 for p ≥ 5.…”

Section: Introductionmentioning

confidence: 99%

On K3 of Witt Vectors of Length Two Over Finite Fields

Geisser¹

1997

K-Theory

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Abstract. We prove that for W 2 (Fq) the Witt vectors of length two over the finite field Fq, we haveThe result is proved by using the identity K 3 (W 2 (Fq)) = H 3 (SL(W 2 (Fq ))) and calculating the right term with a group homology spectral sequence. Some information on the spectral sequence is achieved by using the action of the outer automorphism of SL on the homology groups and recent results on Kgroups of local rings and the ring of dual numbers over finite fields.

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K2 of discrete valuation rings

Cited by 78 publications

References 8 publications

Universal lattices and unbounded rank expanders

Universal lattices and unbounded rank expanders

Universal lattices and property tau

On K3 of Witt Vectors of Length Two Over Finite Fields

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