1 Arithmetic groups and S-arithmetic groups.

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1 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(F n ), MCG, CAT(0)-groups and others. A first example of an arithmetic group is the modular group SL 2 (Z). The action of SL 2 (Z) on the hyperbolic plane H 2 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 primarily 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. 1 Arithmetic groups and S-arithmetic groups. Let K be a global field, a finite extension of Q or F p (t). Given an algebraic group G defined over K (such as SL n ), then G(K) can be viewed as a matrix group with matrix entries in K. Denote the set of inequivalent valuations on K by V K and the set of all inequivalent Archamedian valuations on K by VK. Let S be a finite subset of V K that contains VK. The S-integers of K are O S = {x K : ν(x) 0 ν V K S}. An S-arithmetic group is obtained by restricting the entries of G(K) to a ring of integers O S K and is denoted G(O S ). In the case where S = V K, G(O S) 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 O S of Q is Z. This shows that SL n (O S ) = SL n (Z) is an arithmetic group. This is one of the most prominent examples of an arithmetic group. The group SL n (Z) acts on the symmetric space SO n (R)\SL n (R) (when n = 2 this is the hyperbolic plane). Example 2. Let K = Q( 2) be a degree two extension of Q. 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 SL n (O S ) = SL n (Z[ 2]). The group SL n (Z[ 2]) does not act discretely on the

2 symmetric space for SL n (R) however it does act discretely on a product of symmetric spaces SO n (R)\SL n (R) SO n (R)\SL n (R) one for each embedding of Z[ 2] into R. Example 3. Let K = Q and let S = {ν, ν p }. The valuation ν p is the p-adic valuation given by ν p (x) = n where x = p n a b and p does not divide a or b. Then O S = Z[ 1 p ] and SL 2 (O S ) = SL 2 (Z[ 1 p ]). Note that SL 2(Z[ 1 p ]) is not a discrete subgroup of SL 2(R) or SL 2 (Q p ), however, it is a discrete subgroup of SL 2 (R) SL 2 (Q p ) (via the diagonal embedding). The group consequently acts discretely on H 2 T p - the product of a hyperbolic plane and regular (p + 1)-valent tree. These examples illustrate the general case. For ν S let K ν be the completion of K with respect to the norm induced by ν. If K ν is an Archimedean field, G(K ν ) acts on a symmetric space X ν. If K ν is not an Archimedean field, then G(K ν ) acts on a Euclidean building X ν. In this way a space associated to G = ν S G(K ν ) is taken to be the product: X = v S X v. The group G(O S ) embeds diagonally into G and therefore there is an action of the S-arithmetic group G(O S ) on X - the product of symmetric spaces and Euclidean buildings. 2 Finiteness properties. Definition 1. A group G is said to be of type F m 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 for S-arithmetic groups depend on the sum of the local ranks: Definition 2. For any field extension L/K, the L-rank of G, denoted rank L G, 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) = v S rank Kv G. This number is called the sum of the local ranks and is the same as the dimension of a maximal Euclidean flat in X.

3 Bux-Köhl-Witzel recently proved the Rank Theorem, showing that every S-arithmetic subgroup Γ of a noncommutative K-isotropic absolutely almost simple group G defined over a global function field K is of type F k(g,s) 1 but is not of type F k(g,s) [BKW13]. Applying this theorem to SL n (F p [t]) shows that SL n (F p [t]) is of type F n 2 but not of type F n 1. A main technique in this process is considering the action of Γ on a Euclidean building. The group SL n (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 SL n (Z[t]). This is because SL n (Z[t]) SL n (Q((t 1 ))) and since Q((t 1 )) is a discrete valuation field there is a Euclidean building that SL n (Q((t 1 ))) acts on. If a group G is of type F m, then by definition (using the cellular chain complex of X) there is a free resolution of the trivial ZG-module Z which is finitely generated up to dimension m. This suggests the following weakening of the finiteness condition F m. Definition 3. A group G is of type F P m (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 is F m then it is F P m. In [BB97], Bestvina and Brady give an example to show that F P m is strictly weaker than F m (see example 6.3.3). In [BMW10], Bux-Mohammadi-Wortman make use of action of SL n (Z[t]) on the Euclidean building for SL n (Q((t 1 ))) and Brown s filtration criterion [Bro87] to show that SL n (Z[t]) is not F P n 1. If a finitely generated group G is of type F P m then H m (G; R) is a finitely generated R-module. However, if a group fails to be F P m then it is not necessarily the case that H m (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 G is not F P m. The result of Bux-Mohammadi-Wortman raises the following question. Question 2.1. Is H n 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 of SL n (Z[t]) such that for every Γ in the family of subgroups, H n 1 (Γ; Q) is infinite dimensional [CK13]. Definition 4. Given any ring A and a proper ideal I A, the subgroup SL n (A, I) = ker(sl n (A) SL n (A/I)) is called a principal congruence subgroup of SL n (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 H n 1 (SL n (Z[t], I); Q) is infinite dimensional.

4 The proof of Theorem 2 again makes use of the action of (SL n (Z[t], I) on Euclidean building for SL n (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 H n 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 for SL n (F p [t]) and has shown that there is a finite index subgroup Γ SL n (F p [t]) such that H n 1 (Γ; F p ) is infinite dimensional [Wor13]. Bux-Wortman have used an action of SL 2 (Z[t, t 1 ]) on the product of two buildings to show that SL 2 (Z[t, t 1 ]) is not F P 2 and Cobb has extended this result to give a new proof of Knusdon s theorem that H 2 (SL 2 (Z[t, t 1 ]); Q) is infinite dimensional [BW06], [Cob13], [Knu08]. This work raises the following interesting question: Question 2.2. Is SL n (Z[t, t 1 ]) type F P 2(n 1)? Is H 2(n 1) (SL n (Z[t, t 1 ]; Q) infinite dimensional? It is also of interest to consider other algebraic groups such as the symplectic group Sp 2n. By studying the Euclidean building for Sp 2n (Q((t 1 ))) it is likely one should be able to prove results about finiteness properties of Sp 2n (Z[t]) by generalizing techniques from [BMW10],[CK13], [Cob13],[Kel13a],[Wor13]. Question 2.3. Is Sp 2n (Z[t]) of type F k? Is H k (Sp 2n (Z[t]); Q) an infinite dimensional vector space? 2.1 Solvable Groups For solvable arithmetic groups defined over function fields results about finiteness properties 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(O S ) is of type F S 1 but not type F P S. As a special case, if B n the Borel subgroup of SL n of upper triangular matrices then B n (O S ) is not of type F 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 F p (t). There is a finite index subgroup Γ of B n (O S ) such that, if p 2, the vector space H S (Γ; F p ) is infinite dimensional. Example 4. There is a finite index subgroup Γ of B n (F 3 [t, t 1 ]) such that H 2 (Γ; F 3 ) is infinite dimensional. This theorem is proved by generalizing techniques of [CK13], making use of observations from [Bux04] and looking at the action of B 2 (O S ) on a product of trees.

5 2.2 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 an n-cycle by an (n + 1)-chain. It is a standard result that the diagonal embedding makes the S-arithmetic group G(O S ) a lattice in G. The group G(O S ) acts on X in a natural way via this embedding. The action of G(O S ) on X is not necessarily cocompact. Therefore, as a geometric model for G(O S ) take a closed metric neighborhood of the G(O S ) orbit of some point x. Label this subspace of X as X OS. Given a n-cycle Y X OS, let v XOS (Y ) be the infimum of the volumes of all (n+1)- chains B X OS such that B = Y. Now we can quantify the difficulty of filling an n-cycle by the function R n (G(O S ))(L) = sup{v XOS (Y ) : vol(y ) L}. There are some choices made in the definition of R n (G(O S ))(L) however all these choices do not change the asymptotic class of R n (G(O S )). In general results are stated about the asymptotic class of these filling functions. Question 2.4. For a given S-arithmetic group G(O S ) what are the asymptotics of R n (G(O S ))(L)? Some progress has been made answering this question. If we fix n = 1, this is equivalent to asking about the Dehn function for G(O S ). Examples of non-cocompact arithmetic groups where the Dehn function has been calculated include: SL 2 (Z) linear SL 2 (Z) is hyperbolic SL 3 (Z) exponential Epstein- Thurston [ECH + 92] SL n (Z) ; n > 4 quadratic Young [You13] SL 2 (Z[ 1 ]) p exponential Taback [Tab03] Note that the Dehn function for SL 4 (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 over Q and is Q-isotropic, then R rankr G 1(G(Z)) is bounded below by an exponential function In [Wor11], Wortman contributes the following theorem

6 Theorem 2.3 (Wortman). Let G be a connected semisimple, Q-group that is almost simple over Q. Assume that G is Q-isotopic. Furthermore, suppose the Q-relative root system of G is of type A n, B n, C n, D n, E 6, or E 7. Then there exist constants C > 0 and L 0 > 0 such that R rankr G 1(G(Z))(L) e CL I am currently working on the following conjecture that puts much of what is known into a common language of S-arithmetic groups [Kel13b]. Conjecture 2.2. Let K be a number field and S a finite set of inequivalent valuations on K and let G be a noncommutative, K-isotropic, absolutely almost simple, K-group. There exists a C > 0 and an L 0 > 0 such that R k(g,s) 1 (G(O S ))(L) e CL for all L > L 0. To date I have been able to make progress on Conjecture 2.2 and am able to prove the conjecture so long as G is of type A n, B n, C n, D n, E 6, or E 7 [Kel13b]: In particular this proves that Proposition 2.4 (Kelly). Let Γ = SL n (Z[ 1 p ]). There exist constants C > 0 and L 0 > 0 such that R 2n 3 (Γ)(L) e CL As a special case this proposition includes a new proof that SL 2 (Z[ 1 ]) has exponential Dehn p function. References [BB97] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Inventiones mathematicae 129 (1997), [BKW13] K.-U. Bux, R. Kohl, and S. Witzel, Higher Finiteness Properties of Reductive Arithmetic Groups in Positive Characteristic: The Rank Theorem, Annals of Mathematics 177 (2013), [BMW10] K.-U. Bux, A. Mohammadi, and K. Wortman, SL(n, Z[t]) is not F P n 1, Commentarii Mathematici Helvetici 85 (2010), [Bro87] K. Brown, Finiteness porperties of groups, J. Pure Appl. Algebra 44 (1987), [Bux04] K.-U. Bux, Finiteness properties of soluble arithmetic groups over funcion fields, Geometry Topology 8 (2004),

7 [BW06] [CK13] K.-U. Bux and K. Wortman, A geometric proof that SL n (Z[t, t 1 ]) is not finitely presented, Algebr. Geom. Topol. 6 (2006), M. Cesa and B. Kelly, Congruence subgroups of SL n (Z[t]) with infinite dimensional cohomology, In preperation. (2013). [Cob13] S. Cobb, H 2 (SL 2 Z[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, [Kel13a] [Kel13b] [Knu08] [LP96] B. Kelly, A finite index subgroup of H S (B n (O s ); F p ) with infinite dimensional cohomology., Preprint. (2013)., Higher dimensional isoperimetric functions for non-cocompact arithmetic and S-arithmetic groups, In preperation. (2013). K. Knusdon, Homology and finitenss properties of for SL 2 (Z[t, t 1 ]), Algebr. Geom. Topol. (2008). E. Leuzinger and C. Pittet, Isoperimetric inequalities for lattices in semisimple lie groups of rank 2., Geom. Funct. Anal 6 (1996), [Tab03] J. Taback, The dehn function of P SL(2, Z[1/p]), Geom. Dedicata 102 (2003), [Wor11] [Wor13] K. Wortman, Exponential higher dimensional isoperimetric inequalities for some arithmetic groups, Geom. Dedicata. (2011)., 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 (2013),