We study quadratic functionals on L2(Rd) that generate seminorms in the fractional Sobolev space Hs(Rd) for 0<s<1. The functionals under consideration appear in the study of Markov jump processes ...and, independently, in recent research on the Boltzmann equation. The functional measures differentiability of a function f in a similar way as the seminorm of Hs(Rd). The major difference is that differences f(y)−f(x) are taken into account only if y lies in some double cone with apex at x or vice versa. The configuration of double cones is allowed to be inhomogeneous without any assumption on the spatial regularity. We prove that the resulting seminorm is comparable to the standard one of Hs(Rd). The proof follows from a similar result on discrete quadratic forms in Zd, which is our second main result. We establish a general scheme for discrete approximations of nonlocal quadratic forms. Applications to Markov jump processes are discussed.
Surface Houghton groups Aramayona, Javier; Bux, Kai-Uwe; Kim, Heejoung ...
Mathematische Annalen,
08/2024, Letnik:
389, Številka:
4
Journal Article
Recenzirano
Odprti dostop
For every
n
≥
2
, the
surface Houghton group
B
n
is defined as the asymptotically rigid mapping class group of a surface with exactly
n
ends, all of them non-planar. The groups
B
n
are analogous to, ...and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some
B
n
. As countable mapping class groups of infinite type surfaces, the groups
B
n
lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that
B
n
is of type
F
n
-
1
, but not of type
FP
n
, analogous to the braided Houghton groups.
On the bordification of Outer space Bux, Kai‐Uwe; Smillie, Peter; Vogtmann, Karen
Journal of the London Mathematical Society,
August 2018, 2018-08-00, Letnik:
98, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We give a simple construction of an equivariant deformation retract of Outer space which is homeomorphic to the Bestvina–Feighn bordification. This results in a much easier proof that the ...bordification is (2n‐5)‐connected at infinity, and hence that Out(Fn) is a virtual duality group.
We show that the finiteness length of an S-arithmetic subgroup Γ in a noncommutative isotropic absolutely almost simple group 𝒢 over a global function field is one less than the sum of the local ...ranks of 𝒢 taken over the places in S. This determines the finiteness properties for S-arithmetic sub-groups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr—Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.
The dimension of the Torelli group BESTVINA, MLADEN; BUX, KAI-UWE; MARGALIT, DAN
Journal of the American Mathematical Society,
01/2010, Letnik:
23, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus g \geq 2 is equal to 3g-5. This answers a question of Mess, who proved the lower ...bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be 2g-3. For g \geq 2, we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the ``complex of minimizing cycles'', on which the Torelli group acts.
Let
be an
S
-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group
G
over a global function field
K
. We show that the sum of local ranks of
G
determines the homological ...finiteness properties of
provided the
K
-rank of
G
is 1. This shows that the general upper bound for the finiteness length of
established in an earlier paper is sharp in this case.
The geometric analysis underlying our result determines the connectivity properties of horospheres in thick Euclidean buildings.
Let (ProQuest: Formulae and/or non-USASCII text omitted; see image) be the kernel of the natural map Out(F^sub n^)arrow rightGL^sub n^(). We use combinatorial Morse theory to prove that (ProQuest: ...Formulae and/or non-USASCII text omitted; see image) has an Eilenberg-MacLane space which is (2n-4)-dimensional and that (ProQuest: Formulae and/or non-USASCII text omitted; see image) is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of (ProQuest: Formulae and/or non-USASCII text omitted; see image) is equal to 2n-4 and recovers the result of Krstic-McCool that (ProQuest: Formulae and/or non-USASCII text omitted; see image) is not finitely presented. We also give a new proof of the fact, due to Magnus, that (ProQuest: Formulae and/or non-USASCII text omitted; see image) is finitely generated. PUBLICATION ABSTRACT
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![SLn(ℤ[t]) is not FPn − 1](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/1420-8946_0010-2571/small "SLn(ℤ[t]) is not FPn − 1")