# 1 Arithmetic groups and S-arithmetic groups.

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## Research Statement

I work on questions in geometric group theory by making use of topological methods to better understand finitely generated groups. Arithmetic groups form a particularly rich family of groups that are examples of latices in Lie groups. The theory of arithmetic groups draws from many fields including geometry, number theory, and group theory. Arithmetic groups share common qualities with many other families of groups, Kac-Moody groups, tree lattices, Aut(Fn), MCG, CAT(0)-groups and others.

A first example of an arithmetic group is the modular groupSL2(Z). The action of

SL2(Z) on the hyperbolic planeH2 showcases that even a first example of an arithmetic group has an interesting geometry.

I study the action of arithmetic groups on Euclidean buildings and symmetric spaces to explore questions about the geometry of arithmetic groups. My research is primar-ily focused on finiteness properties and isoperimetric inequalities for arithmetic and S-arithmetic groups, however I am broadly interested in the geometry of buildings, CAT(0)-spaces, boundaries of CAT(0)-spaces, finiteness properties, group cohomology and isoperimetric inequalities.

## -arithmetic groups.

Let K be a global field, a finite extension of Q or Fp(t). Given an algebraic group

G defined over K (such as SLn), then G(K) can be viewed as a matrix group with matrix entries inK.

Denote the set of inequivalent valuations onK byVK and the set of all inequivalent Archamedian valuations on K by VK∞. Let S be a finite subset of VK that contains VK∞. The S-integers of K are

OS ={x∈K :ν(x)≥0 ∀ν∈VK −S}.

An S-arithmetic group is obtained by restricting the entries of G(K) to a ring of integers OS ⊆K and is denoted G(OS). In the case where S =VK∞, G(OS) is called an arithmetic group.

Example 1. Let K =Q and S = {ν∞} where the valuation ν∞ is given by ν∞(x) =

log(|x|) and gives rise to the typical Archamedian norm. The ring of integers OS of

Q is Z. This shows that SLn(OS) = SLn(Z) is an arithmetic group. This is one of

the most prominent examples of an arithmetic group. The group SLn(Z) acts on the

symmetric space SOn(R)\SLn(R) (when n= 2 this is the hyperbolic plane).

Example 2. LetK =Q(√2)be a degree two extension ofQ. There is a set S⊂V

Q( √

2)

such that the ring of S-integers of Q(√2) is the ring Z[√2] = {a+b√2 : a, b ∈ Z}. Then SLn(OS) = SLn(Z[

2]). The group SLn(Z[

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symmetric space for SLn(R) however it does act discretely on a product of symmetric

spaces

SOn(R)\SLn(R)×SOn(R)\SLn(R)

one for each embedding of Z[√2] into R.

Example 3. LetK =Qand letS ={ν∞, νp}. The valuationνp is thep-adic valuation

given by νp(x) =n where x =pn ab and p does not divide a or b. Then OS =Z[1p] and

SL2(OS) = SL2(Z[1p]). Note that SL2(Z[p1]) is not a discrete subgroup of SL2(R) or

SL2(Qp), however, it is a discrete subgroup of SL2(R)×SL2(Qp) (via the diagonal

embedding). The group consequently acts discretely on H2 × T

p - the product of a

hyperbolic plane and regular (p+ 1)-valent tree.

These examples illustrate the general case. For ν ∈S let Kν be the completion of K with respect to the norm induced by ν. If Kν is an Archimedean field, G(Kν) acts on a symmetric space Xν. If Kν is not an Archimedean field, then G(Kν) acts on a Euclidean buildingXν. In this way a space associated to

G=Y ν∈S

G(Kν)

is taken to be the product:

X =Y v∈S

Xv.

The group G(OS) embeds diagonally into G and therefore there is an action of the S-arithmetic group G(OS) on X - the product of symmetric spaces and Euclidean buildings.

## Finiteness properties.

Definition 1. A group G is said to be of type Fm if G acts freely on a contractible CW complex X such that the m-skeleton of G\X is finite.

Many results about finiteness properties forS-arithmetic groups depend on the sum of the local ranks:

Definition 2. For any field extension L/K, the L-rank of G, denoted rankLG, is the

dimension of a maximal L-split torus of G. For any K-group G and set of places S, we define the nonnegative integer

k(G, S) =X v∈S

rankKvG.

This number is called the sum of the local ranks and is the same as the dimension of a maximal Euclidean flat in X.

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Bux-K¨ohl-Witzel recently proved the Rank Theorem, showing that everyS-arithmetic subgroup Γ of a noncommutativeK-isotropic absolutely almost simple groupGdefined over a global function fieldK is of type Fk(G,S)−1 but is not of typeFk(G,S) [BKW13]. Applying this theorem to SLn(Fp[t]) shows that SLn(Fp[t]) is of type Fn−2 but not of type Fn−1. A main technique in this process is considering the action of Γ on a Euclidean building.

The groupSLn(Z[t]) is not an arithmetic group. However, many of the techniques used for arithmetic groups can be employed to gain results about finiteness properties of

SLn(Z[t]). This is because SLn(Z[t])⊆SLn(Q((t−1))) and sinceQ((t−1)) is a discrete valuation field there is a Euclidean building that SLn(Q((t−1))) acts on.

If a group Gis of type Fm, then by definition (using the cellular chain complex of X) there is a free resolution of the trivialZG-module Z which is finitely generated up to dimension m. This suggests the following weakening of the finiteness conditionFm.

Definition 3. A group G is of type F Pm (with respect to Z) if there is a projective

resolution of the trivial ZG-module Z that is finitely generated up to dimension m.

It is clear that if a group isFm then it isF Pm. In [BB97], Bestvina and Brady give an example to show that F Pm is strictly weaker thanFm (see example 6.3.3).

In [BMW10], Bux-Mohammadi-Wortman make use of action of SLn(Z[t]) on the Euclidean building forSLn(Q((t−1))) and Brown’s filtration criterion [Bro87] to show that SLn(Z[t]) is notF Pn−1.

If a finitely generated groupGis of typeF Pm thenHm(G;R) is a finitely generated R-module. However, if a group fails to be F Pm then it is not necessarily the case that Hm(G;R) is an infinitely generated R-module. So asking if H

m(G,−) is finitely generated becomes an interesting question even when we know that Gis not F Pm.

The result of Bux-Mohammadi-Wortman raises the following question.

Question 2.1. Is Hn−1(SL

n(Z[t]);Q) an ifinite dimensional vector space?

I have am working on this question in collaboration with Cesa. Our contribution to this question will be demonstrating that there is a family of subgroups ofSLn(Z[t]) such that for every Γ in the family of subgroups, Hn−1(Γ;

Q) is infinite dimensional

[CK13].

Definition 4. Given any ring A and a proper ideal I ⊆A, the subgroup

SLn(A,I) =ker(SLn(A)→SLn(A/I))

is called a principal congruence subgroup of SLn(A).

Theorem 2.1 (Cesa-Kelly). Fix n ∈ N. Let I ⊂ Z[t] be a nontrivial ideal. If n ≥ 3

then we further assume that I ∩ Z = (0). Then Hn−1(SL

n(Z[t],I);Q) is infinite

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The proof of Theorem 2 again makes use of the action of (SLn(Z[t],I) on Euclidean building forSLn(Q((t−1))) and constructs an explicit family of independent cocycles.

The heart of the argument makes calculations about the cohomology of the link of a vertex (which is a spherical building) and uses an averaging technique to produce a global cocycle on the building.

Theorem 2 does not answer the question of whether Hn−1(SL

n(Z[t]);Q) is infinite dimensional and this remains an open question of interest to me.

Wortman has made progress on an analogous question forSLn(Fp[t]) and has shown that there is a finite index subgroup Γ ≤ SLn(Fp[t]) such that Hn−1(Γ;Fp) is infinite dimensional [Wor13].

Bux-Wortman have used an action ofSL2(Z[t, t−1]) on the product of two buildings to show thatSL2(Z[t, t−1]) is notF P2 and Cobb has extended this result to give a new proof of Knusdon’s theorem thatH2(SL2(Z[t, t−1]);Q) is infinite dimensional [BW06], [Cob13] , [Knu08]. This work raises the following interesting question:

Question 2.2. Is SLn(Z[t, t−1]) type F P2(n−1)? Is H2(n−1)(SLn(Z[t, t−1];Q) infinite

dimensional?

It is also of interest to consider other algebraic groups such as the symplectic group

Sp2n. By studying the Euclidean building forSp2n(Q((t−1))) it is likely one should be able to prove results about finiteness properties ofSp2n(Z[t]) by generalizing techniques from [BMW10],[CK13], [Cob13],[Kel13a],[Wor13].

Question 2.3. IsSp2n(Z[t])of type Fk? IsHk(Sp2n(Z[t]);Q)an infinite dimensional

vector space?

### Solvable Groups

For solvable arithmetic groups defined over function fields results about finiteness prop-erties are not predicated on the the rank of the group. In [Bux04], Bux shows that if G is a Chevalley group and B ≤ G is a Borel subgroup, then B(OS) is of type F|S|−1 but not typeF P|S|. As a special case, ifBn the Borel subgroup ofSLn of upper triangular matrices thenBn(OS) is not of typeF P|S|. I have contributed the following

strengthening of Bux’s theorem in this special case [Kel13a].

Theorem 2.2 (Kelly). Let S be a finite set of discrete valuations on Fp(t). There is

a finite index subgroup Γ of Bn(OS) such that, if p= 26 , the vector space H|S|(Γ;Fp) is infinite dimensional.

Example 4. There is a finite index subgroup Γ of Bn(F3[t, t−1]) such that H2(Γ;F3)

is infinite dimensional.

This theorem is proved by generalizing techniques of [CK13], making use of obser-vations from [Bux04] and looking at the action of B2(OS) on a product of trees.

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### Isoperimetric Inequalities

Filling invariants for a group provide insight to the group’s large scale geometry. The most well studied invariant is the Dehn function, which describes the difficulty of filling a closed curve with a disk. Higher dimensional analogues can be similarly phrased. One path to pursue is to investigate functions that describe the difficulty of filling ann-cycle by an (n+ 1)-chain.

It is a standard result that the diagonal embedding makes the S-arithmetic group

G(OS) a lattice inG. The groupG(OS) acts onX in a natural way via this embedding. The action of G(OS) on X is not necessarily cocompact. Therefore, as a geometric model forG(OS) take a closed metric neighborhood of the G(OS) orbit of some point x. Label this subspace ofX as XOS.

Given an-cycleY ⊆XOS, letvXOS(Y) be the infimum of the volumes of all (n+

1)-chains B ⊆ XOS such that ∂B = Y. Now we can quantify the difficulty of filling an n-cycle by the function

Rn(G(OS))(L) = sup{vXOS(Y) : vol(Y)≤L}.

There are some choices made in the definition of Rn(G(OS))(L) however all these choices do not change the asymptotic class ofRn(G(OS)). In general results are stated about the asymptotic class of these filling functions.

Question 2.4. For a given S-arithmetic group G(OS) what are the asymptotics of Rn(G(OS))(L)?

Some progress has been made answering this question. If we fix n = 1, this is equivalent to asking about the Dehn function forG(OS). Examples of non-cocompact arithmetic groups where the Dehn function has been calculated include:

SL2(Z) linear SL2(Z) is hyperbolic

SL3(Z) exponential Epstein- Thurston [ECH+92]

SLn(Z) ;n >4 quadratic Young [You13]

SL2(Z[1p]) exponential Taback [Tab03]

Note that the Dehn function for SL4(Z) is unknown; Thurston conjectured that it is quadratic.

The following conjecture of Leuzinger-Pittet has informed my work in the area [LP96].

Conjecture 2.1 (Leuzinger-Pittet). If G is a connected, semisimple, Q-group that is almost simple overQ and isQ-isotropic, then RrankRG−1(G(Z))is bounded below by an

exponential function

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Theorem 2.3 (Wortman). Let G be a connected semisimple, Q-group that is almost simple over Q. Assume that GisQ-isotopic. Furthermore, suppose theQ-relative root system ofG is of typeAn, Bn, Cn, Dn, E6,or E7. Then there exist constants C >0 and L0 >0 such that

RrankRG−1(G(Z))(L)≥eCL

I am currently working on the following conjecture that puts much of what is known into a common language ofS-arithmetic groups [Kel13b].

Conjecture 2.2. Let K be a number field andS a finite set of inequivalent valuations onK and letGbe a noncommutative, K-isotropic, absolutely almost simple, K-group. There exists a C > 0 and an L0 >0 such that

Rk(G,S)−1(G(OS))(L)≥eCL for all L > L0.

To date I have been able to make progress on Conjecture 2.2 and am able to prove the conjecture so long asG is of type An, Bn, Cn, Dn, E6, orE7 [Kel13b]:

In particular this proves that

Proposition 2.4(Kelly). LetΓ =SLn(Z[1p]). There exist constantsC >0andL0 >0

such that

R2n−3(Γ)(L)≥eCL

As a special case this proposition includes a new proof that SL2(Z[1p]) has expo-nential Dehn function.

## References

[BB97] M. Bestvina and N. Brady,Morse theory and finiteness properties of groups, Inventiones mathematicae 129 (1997), 445–470.

[BKW13] K.-U. Bux, R. Kohl, and S. Witzel, Higher Finiteness Properties of Re-ductive Arithmetic Groups in Positive Characteristic: The Rank Theorem, Annals of Mathematics 177 (2013), 311–366.

[BMW10] K.-U. Bux, A. Mohammadi, and K. Wortman, SL(n,Z[t]) is not F Pn−1, Commentarii Mathematici Helvetici 85 (2010), 151–164.

[Bro87] K. Brown,Finiteness porperties of groups, J. Pure Appl. Algebra44(1987), 45–75.

[Bux04] K.-U. Bux, Finiteness properties of soluble arithmetic groups over funcion fields, Geometry Topology 8 (2004), 611–644.

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[BW06] K.-U. Bux and K. Wortman, A geometric proof that SLn(Z[t, t−1]) is not

finitely presented, Algebr. Geom. Topol. 6 (2006), 839–852.

[CK13] M. Cesa and B. Kelly, Congruence subgroups of SLn(Z[t]) with infinite

di-mensional cohomology, In preperation. (2013). [Cob13] S. Cobb, H2(SL

2Z[t, t1]);Q) is innite-dimensional, Preprint (2013).

[ECH+92] D.B.A. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, and W. Thurston, Word processing in groups, Jones and Bartlett Publishers, 1992.

[Kel13a] B. Kelly, A finite index subgroup of H|S|(B

n(Os);Fp) with infinite

dimen-sional cohomology., Preprint. (2013).

[Kel13b] , Higher dimensional isoperimetric functions for non-cocompact arithmetic and S-arithmetic groups, In preperation. (2013).

[Knu08] K. Knusdon,Homology and finitenss properties of forSL2(Z[t, t−1]), Algebr. Geom. Topol. (2008).

[LP96] E. Leuzinger and C. Pittet,Isoperimetric inequalities for lattices in semisim-ple lie groups of rank 2., Geom. Funct. Anal 6 (1996), 489–511.

[Tab03] J. Taback, The dehn function of P SL(2,Z[1/p]), Geom. Dedicata 102

(2003), 179 –194.

[Wor11] K. Wortman, Exponential higher dimensional isoperimetric inequalities for some arithmetic groups, Geom. Dedicata. (2011).

[Wor13] , On the cohomology of arithmetic groups over function fields, preprint (2013).

[You13] R. Young, The dehn function of SL(n;Z), Annals of Mathematics 177

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